| L(s) = 1 | − 16·2-s + 56·3-s + 192·4-s − 14·5-s − 896·6-s − 2.04e3·8-s − 1.07e3·9-s + 224·10-s + 2.40e3·11-s + 1.07e4·12-s + 1.07e4·13-s − 784·15-s + 2.04e4·16-s − 3.50e4·17-s + 1.71e4·18-s − 2.40e3·19-s − 2.68e3·20-s − 3.85e4·22-s − 6.16e4·23-s − 1.14e5·24-s + 2.99e4·25-s − 1.71e5·26-s − 1.73e5·27-s − 9.56e4·29-s + 1.25e4·30-s − 1.66e5·31-s − 1.96e5·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.19·3-s + 3/2·4-s − 0.0500·5-s − 1.69·6-s − 1.41·8-s − 0.490·9-s + 0.0708·10-s + 0.545·11-s + 1.79·12-s + 1.35·13-s − 0.0599·15-s + 5/4·16-s − 1.73·17-s + 0.693·18-s − 0.0805·19-s − 0.0751·20-s − 0.771·22-s − 1.05·23-s − 1.69·24-s + 0.382·25-s − 1.91·26-s − 1.69·27-s − 0.728·29-s + 0.0848·30-s − 1.00·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.157933823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.157933823\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 56 T + 1403 p T^{2} - 56 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 14 T - 5941 p T^{2} + 14 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2408 T - 4263503 T^{2} - 2408 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10724 T + 152410814 T^{2} - 10724 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 35098 T + 1018940347 T^{2} + 35098 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2408 T + 1161933513 T^{2} + 2408 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 61684 T + 5919393757 T^{2} + 61684 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 95660 T + 34197541822 T^{2} + 95660 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 166012 T + 60653615117 T^{2} + 166012 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 559814 T + 235867633359 T^{2} + 559814 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 805980 T + 544490570278 T^{2} - 805980 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6232 p T + 534162600534 T^{2} - 6232 p^{8} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1769292 T + 1789746298717 T^{2} - 1769292 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2317194 T + 3230363115583 T^{2} - 2317194 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 660352 T + 4967510319385 T^{2} - 660352 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1463042 T + 5136193527383 T^{2} - 1463042 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1784280 T + 8604082747585 T^{2} + 1784280 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 274400 T - 1862728669394 T^{2} - 274400 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4549062 T + 26670027639019 T^{2} - 4549062 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8673964 T + 50639969018693 T^{2} + 8673964 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10594360 T + 68478299343430 T^{2} + 10594360 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1779750 T + 26770956958699 T^{2} - 1779750 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1748964 T + 72590049947446 T^{2} + 1748964 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.73669312029170055193720093257, −12.15043744021923676757882961180, −11.32475460114429429542670059832, −11.17602873727958656130726069858, −10.61972106196050701948158244371, −9.954107135507584338454003554118, −9.009485056307492176726600151792, −8.960701929093093467703982139109, −8.682966096656110961557473608937, −8.105187144657851268848749814374, −7.29376980566323184818965534265, −6.93663391244607833545606553645, −5.94658560770147008731835373543, −5.67584696866550976714345429075, −3.99441708889574000202854532260, −3.75758279021142527764007846891, −2.58547959850692534262413210468, −2.26240810564051360534339598441, −1.38329015496585012641442087136, −0.39201926283105232149773630032,
0.39201926283105232149773630032, 1.38329015496585012641442087136, 2.26240810564051360534339598441, 2.58547959850692534262413210468, 3.75758279021142527764007846891, 3.99441708889574000202854532260, 5.67584696866550976714345429075, 5.94658560770147008731835373543, 6.93663391244607833545606553645, 7.29376980566323184818965534265, 8.105187144657851268848749814374, 8.682966096656110961557473608937, 8.960701929093093467703982139109, 9.009485056307492176726600151792, 9.954107135507584338454003554118, 10.61972106196050701948158244371, 11.17602873727958656130726069858, 11.32475460114429429542670059832, 12.15043744021923676757882961180, 12.73669312029170055193720093257
