| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s + 25·5-s + 36·6-s − 186.·7-s + 64·8-s + 81·9-s + 100·10-s + 59.2·11-s + 144·12-s + 436.·13-s − 747.·14-s + 225·15-s + 256·16-s + 1.55e3·17-s + 324·18-s − 361·19-s + 400·20-s − 1.68e3·21-s + 236.·22-s − 1.02e3·23-s + 576·24-s + 625·25-s + 1.74e3·26-s + 729·27-s − 2.99e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 1.44·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.147·11-s + 0.288·12-s + 0.715·13-s − 1.01·14-s + 0.258·15-s + 0.250·16-s + 1.30·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.832·21-s + 0.104·22-s − 0.404·23-s + 0.204·24-s + 0.200·25-s + 0.506·26-s + 0.192·27-s − 0.720·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.524090235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.524090235\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 - 9T \) |
| 5 | \( 1 - 25T \) |
| 19 | \( 1 + 361T \) |
| good | 7 | \( 1 + 186.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 59.2T + 1.61e5T^{2} \) |
| 13 | \( 1 - 436.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.55e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.15e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.64e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.13e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.70e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.84e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.26e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.32e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.46e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.55e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 304.T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.13e3T + 8.58e9T^{2} \) |
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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
−9.919908288394782687444685564267, −9.239883844740415309255578394076, −8.150158572342350812055439141174, −7.06137504571524919909805852317, −6.23682616801805290737595608737, −5.49563940223511274685245353359, −4.00555708992001374289002121300, −3.31311463167810370664585286834, −2.34793112079006697157970245399, −0.932142391295291986137767767054,
0.932142391295291986137767767054, 2.34793112079006697157970245399, 3.31311463167810370664585286834, 4.00555708992001374289002121300, 5.49563940223511274685245353359, 6.23682616801805290737595608737, 7.06137504571524919909805852317, 8.150158572342350812055439141174, 9.239883844740415309255578394076, 9.919908288394782687444685564267
