| L(s) = 1 | + 1.96·2-s − 1.75·3-s + 1.87·4-s + 0.290·5-s − 3.44·6-s − 2.03·7-s − 0.249·8-s + 0.0632·9-s + 0.571·10-s − 1.62·11-s − 3.27·12-s − 2.15·13-s − 4.01·14-s − 0.508·15-s − 4.23·16-s − 17-s + 0.124·18-s − 3.82·19-s + 0.543·20-s + 3.56·21-s − 3.19·22-s + 4.47·23-s + 0.437·24-s − 4.91·25-s − 4.24·26-s + 5.13·27-s − 3.81·28-s + ⋯ |
| L(s) = 1 | + 1.39·2-s − 1.01·3-s + 0.936·4-s + 0.129·5-s − 1.40·6-s − 0.770·7-s − 0.0883·8-s + 0.0210·9-s + 0.180·10-s − 0.489·11-s − 0.946·12-s − 0.598·13-s − 1.07·14-s − 0.131·15-s − 1.05·16-s − 0.242·17-s + 0.0293·18-s − 0.878·19-s + 0.121·20-s + 0.778·21-s − 0.681·22-s + 0.933·23-s + 0.0892·24-s − 0.983·25-s − 0.832·26-s + 0.989·27-s − 0.721·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 1.96T + 2T^{2} \) |
| 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 - 0.290T + 5T^{2} \) |
| 7 | \( 1 + 2.03T + 7T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 19 | \( 1 + 3.82T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 3.55T + 29T^{2} \) |
| 31 | \( 1 - 9.03T + 31T^{2} \) |
| 37 | \( 1 - 4.37T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 2.68T + 53T^{2} \) |
| 59 | \( 1 - 8.62T + 59T^{2} \) |
| 61 | \( 1 + 4.69T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 9.13T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 7.63T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−10.20249742475880609980218580251, −9.304802053356928375698902842062, −8.071302888123564264638710379393, −6.57919068349132840376503631849, −6.38122556795597730655589975124, −5.26586651723312163148572646294, −4.75710408721870182009411095392, −3.51167641547133325165950826536, −2.46931529185023526413853953681, 0,
2.46931529185023526413853953681, 3.51167641547133325165950826536, 4.75710408721870182009411095392, 5.26586651723312163148572646294, 6.38122556795597730655589975124, 6.57919068349132840376503631849, 8.071302888123564264638710379393, 9.304802053356928375698902842062, 10.20249742475880609980218580251
