| L(s) = 1 | + 3.21·2-s − 6.73·3-s + 2.31·4-s − 9.24·5-s − 21.6·6-s − 8.56·7-s − 18.2·8-s + 18.4·9-s − 29.6·10-s + 11·11-s − 15.5·12-s − 27.5·14-s + 62.3·15-s − 77.1·16-s + 1.99·17-s + 59.1·18-s + 77.2·19-s − 21.3·20-s + 57.7·21-s + 35.3·22-s + 145.·23-s + 123.·24-s − 39.5·25-s + 57.8·27-s − 19.8·28-s + 131.·29-s + 200.·30-s + ⋯ |
| L(s) = 1 | + 1.13·2-s − 1.29·3-s + 0.289·4-s − 0.826·5-s − 1.47·6-s − 0.462·7-s − 0.806·8-s + 0.682·9-s − 0.938·10-s + 0.301·11-s − 0.375·12-s − 0.525·14-s + 1.07·15-s − 1.20·16-s + 0.0284·17-s + 0.774·18-s + 0.933·19-s − 0.239·20-s + 0.599·21-s + 0.342·22-s + 1.32·23-s + 1.04·24-s − 0.316·25-s + 0.412·27-s − 0.133·28-s + 0.844·29-s + 1.21·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3.21T + 8T^{2} \) |
| 3 | \( 1 + 6.73T + 27T^{2} \) |
| 5 | \( 1 + 9.24T + 125T^{2} \) |
| 7 | \( 1 + 8.56T + 343T^{2} \) |
| 17 | \( 1 - 1.99T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 131.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 312.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 27.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 429.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 394.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 602.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 123.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 795.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 717.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 333.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 153.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 481.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 574.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
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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
−8.432798416288975063659268568909, −7.37417220277196326545450555741, −6.49653688735023701240802714058, −6.04390892885744648146808245870, −5.01027948043577708158162722290, −4.66681900756181212339248256226, −3.59551080219948474088475651902, −2.90069199795245701771655265439, −0.974049981824929338696305542000, 0,
0.974049981824929338696305542000, 2.90069199795245701771655265439, 3.59551080219948474088475651902, 4.66681900756181212339248256226, 5.01027948043577708158162722290, 6.04390892885744648146808245870, 6.49653688735023701240802714058, 7.37417220277196326545450555741, 8.432798416288975063659268568909
