| L(s) = 1 | + 76·3-s + 250·5-s + 796·7-s + 2.36e3·9-s − 560·11-s + 4.47e3·13-s + 1.90e4·15-s + 2.95e4·17-s + 1.95e4·19-s + 6.04e4·21-s + 1.29e5·23-s + 4.68e4·25-s + 8.62e4·27-s − 2.11e5·29-s + 1.65e5·31-s − 4.25e4·33-s + 1.99e5·35-s − 2.92e5·37-s + 3.40e5·39-s − 8.51e5·41-s − 1.97e5·43-s + 5.90e5·45-s − 3.41e4·47-s − 1.05e6·49-s + 2.24e6·51-s − 8.76e5·53-s − 1.40e5·55-s + ⋯ |
| L(s) = 1 | + 1.62·3-s + 0.894·5-s + 0.877·7-s + 1.08·9-s − 0.126·11-s + 0.565·13-s + 1.45·15-s + 1.45·17-s + 0.654·19-s + 1.42·21-s + 2.22·23-s + 3/5·25-s + 0.843·27-s − 1.60·29-s + 0.996·31-s − 0.206·33-s + 0.784·35-s − 0.948·37-s + 0.918·39-s − 1.92·41-s − 0.379·43-s + 0.965·45-s − 0.0479·47-s − 1.27·49-s + 2.36·51-s − 0.808·53-s − 0.113·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.710347683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.710347683\) |
| \(L(\frac{9}{2})\) |
|
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 | $D_{4}$ | \( 1 - 76 T + 1138 p T^{2} - 76 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 796 T + 1687694 T^{2} - 796 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 560 T + 5579446 T^{2} + 560 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4476 T + 88618382 T^{2} - 4476 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 29524 T + 1024708486 T^{2} - 29524 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 19560 T + 450646342 T^{2} - 19560 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 129796 T + 9416757742 T^{2} - 129796 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 211060 T + 35788164814 T^{2} + 211060 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 165352 T + 55844000702 T^{2} - 165352 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 292100 T + 100816677662 T^{2} + 292100 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 20764 p T + 533529345910 T^{2} + 20764 p^{8} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 197812 T + 500337854246 T^{2} + 197812 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 34164 T + 1008165035550 T^{2} + 34164 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 876404 T + 2203297594078 T^{2} + 876404 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2088504 T + 1860966606646 T^{2} + 2088504 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1409708 T + 4803809602382 T^{2} - 1409708 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2882076 T + 14114698855414 T^{2} + 2882076 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9819448 T + 42279153239662 T^{2} - 9819448 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 189404 T + 1441930580342 T^{2} + 189404 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2028496 T + 25840354532958 T^{2} + 2028496 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1393332 T + 43586027298166 T^{2} + 1393332 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 5688820 T + 96481985626742 T^{2} - 5688820 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20045996 T + 226498473606630 T^{2} + 20045996 p^{7} T^{3} + p^{14} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.76430445870802188522027398100, −14.40387693224766381839028461536, −13.69098230988053021748254504218, −13.56042599780776629641516306583, −12.78265179266623581806616031879, −12.05853421660645943182577611861, −11.15686854043810376489146648946, −10.64709360972463975018199306595, −9.624574814641456023898643307164, −9.458095915650258549112104793374, −8.445827874440600787099190818486, −8.285004557704822512179849431553, −7.39815046861630954670907124922, −6.61713521529450674469764129877, −5.39496936619507760428325771391, −4.92101178388324745186976307624, −3.36714152657010877278064898334, −3.06153547827333475651716533639, −1.83513512494041941225325762419, −1.18830893952473561742088446353,
1.18830893952473561742088446353, 1.83513512494041941225325762419, 3.06153547827333475651716533639, 3.36714152657010877278064898334, 4.92101178388324745186976307624, 5.39496936619507760428325771391, 6.61713521529450674469764129877, 7.39815046861630954670907124922, 8.285004557704822512179849431553, 8.445827874440600787099190818486, 9.458095915650258549112104793374, 9.624574814641456023898643307164, 10.64709360972463975018199306595, 11.15686854043810376489146648946, 12.05853421660645943182577611861, 12.78265179266623581806616031879, 13.56042599780776629641516306583, 13.69098230988053021748254504218, 14.40387693224766381839028461536, 14.76430445870802188522027398100
