| L(s) = 1 | − 2-s + 3-s + 4-s − 1.52·5-s − 6-s − 0.354·7-s − 8-s + 9-s + 1.52·10-s − 0.918·11-s + 12-s − 13-s + 0.354·14-s − 1.52·15-s + 16-s + 3.16·17-s − 18-s − 6.07·19-s − 1.52·20-s − 0.354·21-s + 0.918·22-s − 6.90·23-s − 24-s − 2.66·25-s + 26-s + 27-s − 0.354·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.684·5-s − 0.408·6-s − 0.134·7-s − 0.353·8-s + 0.333·9-s + 0.483·10-s − 0.277·11-s + 0.288·12-s − 0.277·13-s + 0.0948·14-s − 0.394·15-s + 0.250·16-s + 0.766·17-s − 0.235·18-s − 1.39·19-s − 0.342·20-s − 0.0774·21-s + 0.195·22-s − 1.43·23-s − 0.204·24-s − 0.532·25-s + 0.196·26-s + 0.192·27-s − 0.0670·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9742650453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9742650453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 1.52T + 5T^{2} \) |
| 7 | \( 1 + 0.354T + 7T^{2} \) |
| 11 | \( 1 + 0.918T + 11T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 + 6.07T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 29 | \( 1 + 4.30T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 - 9.91T + 37T^{2} \) |
| 41 | \( 1 - 1.11T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 9.29T + 47T^{2} \) |
| 53 | \( 1 + 0.486T + 53T^{2} \) |
| 59 | \( 1 - 8.57T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 + 0.566T + 71T^{2} \) |
| 73 | \( 1 + 4.54T + 73T^{2} \) |
| 79 | \( 1 - 17.1T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 7.17T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−7.948465148033507825982381528041, −7.46634158090546891278779370011, −6.59455762215143474852540751467, −5.94373926483719429513918284903, −5.01700676365552108649085351891, −3.94552869822382501254470179764, −3.62221926678662034707548731863, −2.44351999717506066505610495047, −1.89337574854277015991608963399, −0.50512939130915757425733717816,
0.50512939130915757425733717816, 1.89337574854277015991608963399, 2.44351999717506066505610495047, 3.62221926678662034707548731863, 3.94552869822382501254470179764, 5.01700676365552108649085351891, 5.94373926483719429513918284903, 6.59455762215143474852540751467, 7.46634158090546891278779370011, 7.948465148033507825982381528041
