| L(s) = 1 | − 3.06·2-s − 2.01·3-s + 1.40·4-s − 11.9·5-s + 6.17·6-s − 10.6·7-s + 20.2·8-s − 22.9·9-s + 36.6·10-s + 65.4·11-s − 2.82·12-s − 61.8·13-s + 32.6·14-s + 24.0·15-s − 73.2·16-s − 22.8·17-s + 70.3·18-s + 36.6·19-s − 16.7·20-s + 21.4·21-s − 200.·22-s − 143.·23-s − 40.7·24-s + 17.4·25-s + 189.·26-s + 100.·27-s − 14.9·28-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 0.387·3-s + 0.175·4-s − 1.06·5-s + 0.420·6-s − 0.575·7-s + 0.893·8-s − 0.849·9-s + 1.15·10-s + 1.79·11-s − 0.0680·12-s − 1.32·13-s + 0.623·14-s + 0.413·15-s − 1.14·16-s − 0.325·17-s + 0.921·18-s + 0.442·19-s − 0.187·20-s + 0.222·21-s − 1.94·22-s − 1.29·23-s − 0.346·24-s + 0.139·25-s + 1.43·26-s + 0.716·27-s − 0.101·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 3.06T + 8T^{2} \) |
| 3 | \( 1 + 2.01T + 27T^{2} \) |
| 5 | \( 1 + 11.9T + 125T^{2} \) |
| 7 | \( 1 + 10.6T + 343T^{2} \) |
| 11 | \( 1 - 65.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 36.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 143.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 213.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 307.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 83.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 59.1T + 6.89e4T^{2} \) |
| 47 | \( 1 - 201.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 406.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 164.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 150.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 169.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 374.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 465.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 66.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 953.T + 9.12e5T^{2} \) |
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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
−8.563585536230795029815744503172, −7.84045083953244532906437938314, −7.06309324669464265416157897991, −6.41429382714377148363738326605, −5.23431048686464363428537184376, −4.23030919077016360061566520243, −3.54848970439448330645595398975, −2.13560866328841483410819572768, −0.75193275056024517399436821596, 0,
0.75193275056024517399436821596, 2.13560866328841483410819572768, 3.54848970439448330645595398975, 4.23030919077016360061566520243, 5.23431048686464363428537184376, 6.41429382714377148363738326605, 7.06309324669464265416157897991, 7.84045083953244532906437938314, 8.563585536230795029815744503172
