| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 81·5-s − 36·6-s − 49·7-s − 64·8-s + 81·9-s − 324·10-s + 191·11-s + 144·12-s − 169·13-s + 196·14-s + 729·15-s + 256·16-s − 871·17-s − 324·18-s − 479·19-s + 1.29e3·20-s − 441·21-s − 764·22-s + 1.38e3·23-s − 576·24-s + 3.43e3·25-s + 676·26-s + 729·27-s − 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.44·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.02·10-s + 0.475·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.836·15-s + 1/4·16-s − 0.730·17-s − 0.235·18-s − 0.304·19-s + 0.724·20-s − 0.218·21-s − 0.336·22-s + 0.546·23-s − 0.204·24-s + 1.09·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.628888283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.628888283\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 81 T + p^{5} T^{2} \) |
| 11 | \( 1 - 191 T + p^{5} T^{2} \) |
| 17 | \( 1 + 871 T + p^{5} T^{2} \) |
| 19 | \( 1 + 479 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1387 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5295 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5940 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13543 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9464 T + p^{5} T^{2} \) |
| 43 | \( 1 - 17387 T + p^{5} T^{2} \) |
| 47 | \( 1 + 8112 T + p^{5} T^{2} \) |
| 53 | \( 1 - 18038 T + p^{5} T^{2} \) |
| 59 | \( 1 - 28784 T + p^{5} T^{2} \) |
| 61 | \( 1 - 14773 T + p^{5} T^{2} \) |
| 67 | \( 1 + 54354 T + p^{5} T^{2} \) |
| 71 | \( 1 - 64608 T + p^{5} T^{2} \) |
| 73 | \( 1 - 39461 T + p^{5} T^{2} \) |
| 79 | \( 1 + 95554 T + p^{5} T^{2} \) |
| 83 | \( 1 + 69634 T + p^{5} T^{2} \) |
| 89 | \( 1 + 51906 T + p^{5} T^{2} \) |
| 97 | \( 1 - 162654 T + p^{5} T^{2} \) |
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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
−9.682043839969212839752049855829, −9.340579746985257965026867978795, −8.508820653170918059108345529054, −7.33572658058923665751512114192, −6.46389775620545961456139240223, −5.68949451960732118054349601943, −4.25543213601089819050079732899, −2.75828248314972333790553340551, −2.05675544531230108374108981300, −0.882814938169230378092230621802,
0.882814938169230378092230621802, 2.05675544531230108374108981300, 2.75828248314972333790553340551, 4.25543213601089819050079732899, 5.68949451960732118054349601943, 6.46389775620545961456139240223, 7.33572658058923665751512114192, 8.508820653170918059108345529054, 9.340579746985257965026867978795, 9.682043839969212839752049855829
