| L(s) = 1 | − 8·3-s + 250·5-s − 1.28e3·7-s − 4.14e3·9-s + 2.94e3·11-s + 2.02e3·13-s − 2.00e3·15-s − 2.76e4·17-s + 5.34e4·19-s + 1.03e4·21-s + 4.21e4·23-s + 4.68e4·25-s + 4.92e4·27-s + 1.43e5·29-s − 2.55e5·31-s − 2.35e4·33-s − 3.22e5·35-s − 3.24e5·37-s − 1.62e4·39-s − 2.57e5·41-s + 5.94e5·43-s − 1.03e6·45-s + 2.16e6·47-s + 6.32e5·49-s + 2.21e5·51-s + 8.49e5·53-s + 7.36e5·55-s + ⋯ |
| L(s) = 1 | − 0.171·3-s + 0.894·5-s − 1.41·7-s − 1.89·9-s + 0.666·11-s + 0.256·13-s − 0.153·15-s − 1.36·17-s + 1.78·19-s + 0.242·21-s + 0.721·23-s + 3/5·25-s + 0.481·27-s + 1.09·29-s − 1.54·31-s − 0.114·33-s − 1.26·35-s − 1.05·37-s − 0.0437·39-s − 0.582·41-s + 1.13·43-s − 1.69·45-s + 3.03·47-s + 0.767·49-s + 0.233·51-s + 0.783·53-s + 0.596·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.520804447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.520804447\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 8 T + 1402 p T^{2} + 8 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 184 p T + 1026822 T^{2} + 184 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2944 T + 23912102 T^{2} - 2944 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 12 p^{2} T - 18566866 T^{2} - 12 p^{9} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 27644 T + 1003140326 T^{2} + 27644 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 53488 T + 2419893558 T^{2} - 53488 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 42120 T + 1988603110 T^{2} - 42120 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 143228 T + 23837125550 T^{2} - 143228 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 255664 T + 49689334302 T^{2} + 255664 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 324164 T + 44114242206 T^{2} + 324164 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 257180 T + 239116486262 T^{2} + 257180 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 594296 T + 547062034782 T^{2} - 594296 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2160920 T + 2179210856630 T^{2} - 2160920 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 849244 T + 2527962442814 T^{2} - 849244 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2244720 T + 5847312305638 T^{2} - 2244720 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 116332 T + 1428444411342 T^{2} - 116332 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4929384 T + 17911553086574 T^{2} - 4929384 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3456432 T + 4024210501582 T^{2} - 3456432 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1393940 T + 18822156961494 T^{2} - 1393940 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 11340512 T + 63037703406558 T^{2} - 11340512 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 7882552 T + 67314163208654 T^{2} - 7882552 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6735180 T + 93023747263222 T^{2} + 6735180 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 23786180 T + 282313784137926 T^{2} - 23786180 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.94230223818644626789263226878, −11.12050113124141722493882846166, −10.98449677595268837288978943444, −10.33720284885311217526747564456, −9.612726541307450414253156564859, −9.289265618191498723424342070567, −8.782517413699981881918602286824, −8.636103232279261340394421820282, −7.48322020919469018376814160591, −6.95370902964656123155334570151, −6.37367893548619844431121771417, −6.07356314670372465652973827728, −5.22287481045115539944350792731, −5.20763659230917582869282226161, −3.60197117069321507165525603343, −3.59640542599729596194325767664, −2.46640411967943606871865069065, −2.35933225683090014057711539257, −0.901792386252552573604631263762, −0.54600450843652568036856585310,
0.54600450843652568036856585310, 0.901792386252552573604631263762, 2.35933225683090014057711539257, 2.46640411967943606871865069065, 3.59640542599729596194325767664, 3.60197117069321507165525603343, 5.20763659230917582869282226161, 5.22287481045115539944350792731, 6.07356314670372465652973827728, 6.37367893548619844431121771417, 6.95370902964656123155334570151, 7.48322020919469018376814160591, 8.636103232279261340394421820282, 8.782517413699981881918602286824, 9.289265618191498723424342070567, 9.612726541307450414253156564859, 10.33720284885311217526747564456, 10.98449677595268837288978943444, 11.12050113124141722493882846166, 11.94230223818644626789263226878
