| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s + 36·6-s + 49·7-s + 64·8-s + 81·9-s + 272·11-s + 144·12-s − 114·13-s + 196·14-s + 256·16-s + 1.02e3·17-s + 324·18-s + 960·19-s + 441·21-s + 1.08e3·22-s + 1.29e3·23-s + 576·24-s − 456·26-s + 729·27-s + 784·28-s − 1.05e3·29-s − 4.34e3·31-s + 1.02e3·32-s + 2.44e3·33-s + 4.08e3·34-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 + 0.677·11-s + 0.288·12-s − 0.187·13-s + 0.267·14-s + 1/4·16-s + 0.857·17-s + 0.235·18-s + 0.610·19-s + 0.218·21-s + 0.479·22-s + 0.510·23-s + 0.204·24-s − 0.132·26-s + 0.192·27-s + 0.188·28-s − 0.231·29-s − 0.812·31-s + 0.176·32-s + 0.391·33-s + 0.606·34-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
| \(L(3)\) |
\(\approx\) |
\(5.843015153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.843015153\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 272 T + p^{5} T^{2} \) |
| 13 | \( 1 + 114 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1022 T + p^{5} T^{2} \) |
| 19 | \( 1 - 960 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1296 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1050 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4348 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5842 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9322 T + p^{5} T^{2} \) |
| 43 | \( 1 - 1196 T + p^{5} T^{2} \) |
| 47 | \( 1 + 20848 T + p^{5} T^{2} \) |
| 53 | \( 1 + 28274 T + p^{5} T^{2} \) |
| 59 | \( 1 - 28340 T + p^{5} T^{2} \) |
| 61 | \( 1 - 23782 T + p^{5} T^{2} \) |
| 67 | \( 1 + 15108 T + p^{5} T^{2} \) |
| 71 | \( 1 - 49372 T + p^{5} T^{2} \) |
| 73 | \( 1 - 23986 T + p^{5} T^{2} \) |
| 79 | \( 1 - 50320 T + p^{5} T^{2} \) |
| 83 | \( 1 + 63364 T + p^{5} T^{2} \) |
| 89 | \( 1 - 2090 T + p^{5} T^{2} \) |
| 97 | \( 1 - 43282 T + p^{5} T^{2} \) |
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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
−9.274624803914510094167062482147, −8.199123387584272270698181381901, −7.50725223291843873557682182016, −6.67573988676101170328426554711, −5.64440180238694822323436847599, −4.80388701122030490580130709363, −3.82212997687667749371432553508, −3.06417774393786203950432178906, −1.94247818510424598472604437587, −0.960459721350508970304389457328,
0.960459721350508970304389457328, 1.94247818510424598472604437587, 3.06417774393786203950432178906, 3.82212997687667749371432553508, 4.80388701122030490580130709363, 5.64440180238694822323436847599, 6.67573988676101170328426554711, 7.50725223291843873557682182016, 8.199123387584272270698181381901, 9.274624803914510094167062482147
