| L(s) = 1 | − 27·3-s − 270·5-s + 729·9-s − 5.72e3·11-s + 4.57e3·13-s + 7.29e3·15-s + 3.65e4·17-s − 5.17e4·19-s + 2.22e4·23-s − 5.22e3·25-s − 1.96e4·27-s − 1.57e5·29-s + 1.03e5·31-s + 1.54e5·33-s − 9.48e4·37-s − 1.23e5·39-s − 6.59e5·41-s − 7.57e4·43-s − 1.96e5·45-s − 4.05e5·47-s − 9.87e5·51-s − 1.34e6·53-s + 1.54e6·55-s + 1.39e6·57-s + 1.30e6·59-s − 1.83e6·61-s − 1.23e6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.965·5-s + 1/3·9-s − 1.29·11-s + 0.576·13-s + 0.557·15-s + 1.80·17-s − 1.73·19-s + 0.381·23-s − 0.0668·25-s − 0.192·27-s − 1.19·29-s + 0.626·31-s + 0.748·33-s − 0.307·37-s − 0.333·39-s − 1.49·41-s − 0.145·43-s − 0.321·45-s − 0.569·47-s − 1.04·51-s − 1.24·53-s + 1.25·55-s + 0.999·57-s + 0.826·59-s − 1.03·61-s − 0.557·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5620804469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5620804469\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 54 p T + p^{7} T^{2} \) |
| 11 | \( 1 + 5724 T + p^{7} T^{2} \) |
| 13 | \( 1 - 4570 T + p^{7} T^{2} \) |
| 17 | \( 1 - 36558 T + p^{7} T^{2} \) |
| 19 | \( 1 + 51740 T + p^{7} T^{2} \) |
| 23 | \( 1 - 22248 T + p^{7} T^{2} \) |
| 29 | \( 1 + 157194 T + p^{7} T^{2} \) |
| 31 | \( 1 - 103936 T + p^{7} T^{2} \) |
| 37 | \( 1 + 94834 T + p^{7} T^{2} \) |
| 41 | \( 1 + 659610 T + p^{7} T^{2} \) |
| 43 | \( 1 + 75772 T + p^{7} T^{2} \) |
| 47 | \( 1 + 405648 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1346274 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1303884 T + p^{7} T^{2} \) |
| 61 | \( 1 + 30062 p T + p^{7} T^{2} \) |
| 67 | \( 1 - 1369388 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2714040 T + p^{7} T^{2} \) |
| 73 | \( 1 + 2868794 T + p^{7} T^{2} \) |
| 79 | \( 1 + 1129648 T + p^{7} T^{2} \) |
| 83 | \( 1 + 5912028 T + p^{7} T^{2} \) |
| 89 | \( 1 - 897750 T + p^{7} T^{2} \) |
| 97 | \( 1 + 13719074 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.777254028025260731138041143412, −8.364557859619011397698700127119, −7.923909889919381381057099825520, −6.94782598527903539756639887723, −5.86168777239148115072567043020, −5.04343624842834481483610872404, −3.99067079430159181189508306627, −3.07713540371094686744572281319, −1.63498833931706212873109233402, −0.32835464561306049869181808523,
0.32835464561306049869181808523, 1.63498833931706212873109233402, 3.07713540371094686744572281319, 3.99067079430159181189508306627, 5.04343624842834481483610872404, 5.86168777239148115072567043020, 6.94782598527903539756639887723, 7.923909889919381381057099825520, 8.364557859619011397698700127119, 9.777254028025260731138041143412
