| L(s) = 1 | + 4·3-s + 5·5-s − 4·7-s + 10·9-s + 5·11-s + 7·13-s + 20·15-s + 12·17-s − 3·19-s − 16·21-s + 4·23-s + 6·25-s + 20·27-s + 6·29-s − 4·31-s + 20·33-s − 20·35-s + 20·37-s + 28·39-s + 3·41-s − 9·43-s + 50·45-s − 7·47-s + 10·49-s + 48·51-s − 6·53-s + 25·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2.23·5-s − 1.51·7-s + 10/3·9-s + 1.50·11-s + 1.94·13-s + 5.16·15-s + 2.91·17-s − 0.688·19-s − 3.49·21-s + 0.834·23-s + 6/5·25-s + 3.84·27-s + 1.11·29-s − 0.718·31-s + 3.48·33-s − 3.38·35-s + 3.28·37-s + 4.48·39-s + 0.468·41-s − 1.37·43-s + 7.45·45-s − 1.02·47-s + 10/7·49-s + 6.72·51-s − 0.824·53-s + 3.37·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(75.46293503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(75.46293503\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 23 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - p T + 19 T^{2} - 61 T^{3} + 148 T^{4} - 61 p T^{5} + 19 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 5 T + 35 T^{2} - 100 T^{3} + 460 T^{4} - 100 p T^{5} + 35 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 7 T + 41 T^{2} - 121 T^{3} + 492 T^{4} - 121 p T^{5} + 41 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 12 T + 105 T^{2} - 602 T^{3} + 2888 T^{4} - 602 p T^{5} + 105 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 3 T + 39 T^{2} + 120 T^{3} + 812 T^{4} + 120 p T^{5} + 39 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 6 T + 85 T^{2} - 258 T^{3} + 2812 T^{4} - 258 p T^{5} + 85 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 4 T + 113 T^{2} + 326 T^{3} + 5068 T^{4} + 326 p T^{5} + 113 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 20 T + 239 T^{2} - 1940 T^{3} + 13080 T^{4} - 1940 p T^{5} + 239 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 3 T + 137 T^{2} - 298 T^{3} + 7838 T^{4} - 298 p T^{5} + 137 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 9 T + 133 T^{2} + 977 T^{3} + 8012 T^{4} + 977 p T^{5} + 133 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 7 T + 194 T^{2} + 959 T^{3} + 13786 T^{4} + 959 p T^{5} + 194 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 6 T + 144 T^{2} + 705 T^{3} + 9444 T^{4} + 705 p T^{5} + 144 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 2 T - 42 T^{2} - 91 T^{3} + 5646 T^{4} - 91 p T^{5} - 42 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 24 T + 416 T^{2} - 4583 T^{3} + 42024 T^{4} - 4583 p T^{5} + 416 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + T + 127 T^{2} + 225 T^{3} + 12248 T^{4} + 225 p T^{5} + 127 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 17 T + 363 T^{2} - 3749 T^{3} + 41528 T^{4} - 3749 p T^{5} + 363 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 16 T + 197 T^{2} - 1234 T^{3} + 11448 T^{4} - 1234 p T^{5} + 197 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 10 T + 299 T^{2} - 2156 T^{3} + 34888 T^{4} - 2156 p T^{5} + 299 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 8 T + 137 T^{2} - 458 T^{3} + 1708 T^{4} - 458 p T^{5} + 137 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 3 T + 243 T^{2} + 1181 T^{3} + 27320 T^{4} + 1181 p T^{5} + 243 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 2 T + 301 T^{2} + 226 T^{3} + 39580 T^{4} + 226 p T^{5} + 301 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.68293154991321636868551624141, −5.25671111263647621753952227508, −5.10244211650645082632408141066, −5.08364116974471925242948343733, −4.96944360848015964159435328149, −4.39351093953093536585607861061, −4.11328716262423956283508809622, −4.08617282766284141728491862273, −3.94081253444707000488836023952, −3.71933069123949817703604367639, −3.59272544265199543250383247470, −3.46110553243350783279586818043, −3.18050328037875543113774941949, −2.93678935804422257333683376019, −2.70009586894463871053497662549, −2.67615953343853287785242526823, −2.62769760809041027383689312557, −1.97009030097406763229838553370, −1.85614530091720609714460312718, −1.76971826873583203707482661381, −1.68393464939281468270442384499, −1.04720123295918110502710195734, −1.03031537725846140172820949525, −0.873979765283231536587868415505, −0.61090269437896549514952388492,
0.61090269437896549514952388492, 0.873979765283231536587868415505, 1.03031537725846140172820949525, 1.04720123295918110502710195734, 1.68393464939281468270442384499, 1.76971826873583203707482661381, 1.85614530091720609714460312718, 1.97009030097406763229838553370, 2.62769760809041027383689312557, 2.67615953343853287785242526823, 2.70009586894463871053497662549, 2.93678935804422257333683376019, 3.18050328037875543113774941949, 3.46110553243350783279586818043, 3.59272544265199543250383247470, 3.71933069123949817703604367639, 3.94081253444707000488836023952, 4.08617282766284141728491862273, 4.11328716262423956283508809622, 4.39351093953093536585607861061, 4.96944360848015964159435328149, 5.08364116974471925242948343733, 5.10244211650645082632408141066, 5.25671111263647621753952227508, 5.68293154991321636868551624141
