| L(s) = 1 | + 2-s − 3-s + 4-s + 4.34·5-s − 6-s − 7-s + 8-s + 9-s + 4.34·10-s − 5.19·11-s − 12-s + 1.61·13-s − 14-s − 4.34·15-s + 16-s − 5.89·17-s + 18-s − 8.17·19-s + 4.34·20-s + 21-s − 5.19·22-s − 2.11·23-s − 24-s + 13.8·25-s + 1.61·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.94·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.37·10-s − 1.56·11-s − 0.288·12-s + 0.447·13-s − 0.267·14-s − 1.12·15-s + 0.250·16-s − 1.42·17-s + 0.235·18-s − 1.87·19-s + 0.970·20-s + 0.218·21-s − 1.10·22-s − 0.441·23-s − 0.204·24-s + 2.76·25-s + 0.316·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 5 | \( 1 - 4.34T + 5T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 + 5.89T + 17T^{2} \) |
| 19 | \( 1 + 8.17T + 19T^{2} \) |
| 23 | \( 1 + 2.11T + 23T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 - 9.30T + 31T^{2} \) |
| 37 | \( 1 + 7.82T + 37T^{2} \) |
| 41 | \( 1 + 5.82T + 41T^{2} \) |
| 43 | \( 1 - 4.33T + 43T^{2} \) |
| 47 | \( 1 + 0.173T + 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 8.87T + 71T^{2} \) |
| 73 | \( 1 - 3.51T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 + 6.10T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−7.03964846102293201109733147176, −6.49603360169091872046442019786, −6.08756839471223656647036823682, −5.45726536434778029657067047926, −4.85543050989196998452691790855, −4.15501757358443577075712340494, −2.82320580295407370151756377820, −2.31968982643531341648278160746, −1.61405266655710419715117312468, 0,
1.61405266655710419715117312468, 2.31968982643531341648278160746, 2.82320580295407370151756377820, 4.15501757358443577075712340494, 4.85543050989196998452691790855, 5.45726536434778029657067047926, 6.08756839471223656647036823682, 6.49603360169091872046442019786, 7.03964846102293201109733147176
