| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 36·6-s + 49·7-s − 64·8-s + 81·9-s − 254·11-s + 144·12-s − 84·13-s − 196·14-s + 256·16-s − 1.47e3·17-s − 324·18-s + 486·19-s + 441·21-s + 1.01e3·22-s + 1.36e3·23-s − 576·24-s + 336·26-s + 729·27-s + 784·28-s + 6.55e3·29-s + 3.53e3·31-s − 1.02e3·32-s − 2.28e3·33-s + 5.91e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·11-s + 0.288·12-s − 0.137·13-s − 0.267·14-s + 1/4·16-s − 1.24·17-s − 0.235·18-s + 0.308·19-s + 0.218·21-s + 0.447·22-s + 0.539·23-s − 0.204·24-s + 0.0974·26-s + 0.192·27-s + 0.188·28-s + 1.44·29-s + 0.659·31-s − 0.176·32-s − 0.365·33-s + 0.877·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 11 | \( 1 + 254 T + p^{5} T^{2} \) |
| 13 | \( 1 + 84 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1478 T + p^{5} T^{2} \) |
| 19 | \( 1 - 486 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1368 T + p^{5} T^{2} \) |
| 29 | \( 1 - 6558 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3530 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7730 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2326 T + p^{5} T^{2} \) |
| 43 | \( 1 - 3476 T + p^{5} T^{2} \) |
| 47 | \( 1 + 28040 T + p^{5} T^{2} \) |
| 53 | \( 1 + 7684 T + p^{5} T^{2} \) |
| 59 | \( 1 - 27304 T + p^{5} T^{2} \) |
| 61 | \( 1 - 30886 T + p^{5} T^{2} \) |
| 67 | \( 1 - 32484 T + p^{5} T^{2} \) |
| 71 | \( 1 + 24262 T + p^{5} T^{2} \) |
| 73 | \( 1 + 29540 T + p^{5} T^{2} \) |
| 79 | \( 1 + 12920 T + p^{5} T^{2} \) |
| 83 | \( 1 + 82364 T + p^{5} T^{2} \) |
| 89 | \( 1 + 90026 T + p^{5} T^{2} \) |
| 97 | \( 1 + 51704 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.502467256931542731874266077087, −8.310277483046232358349680290258, −7.17941897666597570958958514025, −6.58070122411115647231241943195, −5.27458391140657042631339631332, −4.39234803042592926700091366475, −3.08005676554161252010832593744, −2.29740197004577988016093150956, −1.23269327065709962239598798688, 0,
1.23269327065709962239598798688, 2.29740197004577988016093150956, 3.08005676554161252010832593744, 4.39234803042592926700091366475, 5.27458391140657042631339631332, 6.58070122411115647231241943195, 7.17941897666597570958958514025, 8.310277483046232358349680290258, 8.502467256931542731874266077087
