| L(s) = 1 | − 2·3-s − 4-s + 2·5-s + 2·7-s − 2·8-s − 3·9-s − 2·11-s + 2·12-s + 8·13-s − 4·15-s − 16-s − 17-s + 6·19-s − 2·20-s − 4·21-s − 8·23-s + 4·24-s − 7·25-s + 7·27-s − 2·28-s + 14·29-s + 4·32-s + 4·33-s + 4·35-s + 3·36-s + 18·37-s − 16·39-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 0.894·5-s + 0.755·7-s − 0.707·8-s − 9-s − 0.603·11-s + 0.577·12-s + 2.21·13-s − 1.03·15-s − 1/4·16-s − 0.242·17-s + 1.37·19-s − 0.447·20-s − 0.872·21-s − 1.66·23-s + 0.816·24-s − 7/5·25-s + 1.34·27-s − 0.377·28-s + 2.59·29-s + 0.707·32-s + 0.696·33-s + 0.676·35-s + 1/2·36-s + 2.95·37-s − 2.56·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(59^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(59^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3581361327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3581361327\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 59 | $C_1$ | \( ( 1 - T )^{5} \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 + T^{2} + p T^{3} + p T^{4} + p^{2} T^{6} + p^{3} T^{7} + p^{3} T^{8} + p^{5} T^{10} \) |
| 3 | $C_2 \wr S_5$ | \( 1 + 2 T + 7 T^{2} + 13 T^{3} + 31 T^{4} + 41 T^{5} + 31 p T^{6} + 13 p^{2} T^{7} + 7 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 - 2 T + 11 T^{2} - 17 T^{3} + 59 T^{4} - 69 T^{5} + 59 p T^{6} - 17 p^{2} T^{7} + 11 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - 2 T + 19 T^{2} - 13 T^{3} + 167 T^{4} - 57 T^{5} + 167 p T^{6} - 13 p^{2} T^{7} + 19 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 + 2 T + 31 T^{2} + 64 T^{3} + 546 T^{4} + 860 T^{5} + 546 p T^{6} + 64 p^{2} T^{7} + 31 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - 8 T + 5 p T^{2} - 328 T^{3} + 1642 T^{4} - 6048 T^{5} + 1642 p T^{6} - 328 p^{2} T^{7} + 5 p^{4} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 + T + 40 T^{2} - 13 T^{3} + 819 T^{4} - 608 T^{5} + 819 p T^{6} - 13 p^{2} T^{7} + 40 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 - 6 T + 67 T^{2} - 239 T^{3} + 1847 T^{4} - 5219 T^{5} + 1847 p T^{6} - 239 p^{2} T^{7} + 67 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 + 8 T + 5 p T^{2} + 648 T^{3} + 5178 T^{4} + 21312 T^{5} + 5178 p T^{6} + 648 p^{2} T^{7} + 5 p^{4} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 14 T + 155 T^{2} - 1235 T^{3} + 8795 T^{4} - 49839 T^{5} + 8795 p T^{6} - 1235 p^{2} T^{7} + 155 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 + 39 T^{2} + 56 T^{3} + 102 T^{4} + 3728 T^{5} + 102 p T^{6} + 56 p^{2} T^{7} + 39 p^{3} T^{8} + p^{5} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 - 18 T + 265 T^{2} - 2600 T^{3} + 594 p T^{4} - 143084 T^{5} + 594 p^{2} T^{6} - 2600 p^{2} T^{7} + 265 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 10 T + 135 T^{2} + 947 T^{3} + 8107 T^{4} + 44251 T^{5} + 8107 p T^{6} + 947 p^{2} T^{7} + 135 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 + 4 T + 187 T^{2} + 632 T^{3} + 15134 T^{4} + 39432 T^{5} + 15134 p T^{6} + 632 p^{2} T^{7} + 187 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 + 20 T + 359 T^{2} + 3952 T^{3} + 39254 T^{4} + 282872 T^{5} + 39254 p T^{6} + 3952 p^{2} T^{7} + 359 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 10 T + 243 T^{2} + 2043 T^{3} + 24683 T^{4} + 160451 T^{5} + 24683 p T^{6} + 2043 p^{2} T^{7} + 243 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 22 T + 361 T^{2} - 4000 T^{3} + 39010 T^{4} - 313204 T^{5} + 39010 p T^{6} - 4000 p^{2} T^{7} + 361 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 + 147 T^{2} - 200 T^{3} + 12574 T^{4} - 35696 T^{5} + 12574 p T^{6} - 200 p^{2} T^{7} + 147 p^{3} T^{8} + p^{5} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 - 3 T + 278 T^{2} - 837 T^{3} + 35705 T^{4} - 85184 T^{5} + 35705 p T^{6} - 837 p^{2} T^{7} + 278 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 8 T + 245 T^{2} + 1208 T^{3} + 26674 T^{4} + 101056 T^{5} + 26674 p T^{6} + 1208 p^{2} T^{7} + 245 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 10 T + 335 T^{2} - 2155 T^{3} + 44559 T^{4} - 211747 T^{5} + 44559 p T^{6} - 2155 p^{2} T^{7} + 335 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 6 T + 155 T^{2} - 1144 T^{3} + 19446 T^{4} - 78084 T^{5} + 19446 p T^{6} - 1144 p^{2} T^{7} + 155 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 10 T + 365 T^{2} - 2960 T^{3} + 59834 T^{4} - 370444 T^{5} + 59834 p T^{6} - 2960 p^{2} T^{7} + 365 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 + 22 T + 545 T^{2} + 7960 T^{3} + 111198 T^{4} + 1132900 T^{5} + 111198 p T^{6} + 7960 p^{2} T^{7} + 545 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.894068549906690274672122077886, −9.609222559422130271486833946532, −9.452033658188684003112243683075, −9.088972290477939400682165685728, −8.734786658720098999832941415697, −8.295252893982088291039475633356, −8.200615332779697896344993796642, −8.149795908337185927068481967060, −8.025258171528788636961514462059, −7.60458204573274953377151742717, −6.81488755576059906160369376109, −6.61518458668388613761713961064, −6.32334928660076065391026695380, −6.21184657733260530828940271702, −5.94452195117560837024720157517, −5.69670103008811537522652859438, −5.28906722831942976758576939837, −5.11471649722704245802924848991, −4.67620821870832646968177096200, −4.45358248517884431708091819698, −3.58771324009033032494147405224, −3.57516083774820585877572923465, −2.92810026120442141525159730375, −2.37759514072270104676782844303, −1.52770222633640175259453113545,
1.52770222633640175259453113545, 2.37759514072270104676782844303, 2.92810026120442141525159730375, 3.57516083774820585877572923465, 3.58771324009033032494147405224, 4.45358248517884431708091819698, 4.67620821870832646968177096200, 5.11471649722704245802924848991, 5.28906722831942976758576939837, 5.69670103008811537522652859438, 5.94452195117560837024720157517, 6.21184657733260530828940271702, 6.32334928660076065391026695380, 6.61518458668388613761713961064, 6.81488755576059906160369376109, 7.60458204573274953377151742717, 8.025258171528788636961514462059, 8.149795908337185927068481967060, 8.200615332779697896344993796642, 8.295252893982088291039475633356, 8.734786658720098999832941415697, 9.088972290477939400682165685728, 9.452033658188684003112243683075, 9.609222559422130271486833946532, 9.894068549906690274672122077886
