| L(s) = 1 | + 0.0323·2-s − 6.42·3-s − 7.99·4-s − 0.207·6-s + 14.0·7-s − 0.517·8-s + 14.2·9-s − 55.5·11-s + 51.3·12-s + 18.3·13-s + 0.453·14-s + 63.9·16-s − 10.0·17-s + 0.460·18-s + 161.·19-s − 89.9·21-s − 1.79·22-s + 23·23-s + 3.32·24-s + 0.592·26-s + 81.9·27-s − 112.·28-s + 183.·29-s − 144.·31-s + 6.21·32-s + 356.·33-s − 0.324·34-s + ⋯ |
| L(s) = 1 | + 0.0114·2-s − 1.23·3-s − 0.999·4-s − 0.0141·6-s + 0.756·7-s − 0.0228·8-s + 0.527·9-s − 1.52·11-s + 1.23·12-s + 0.390·13-s + 0.00865·14-s + 0.999·16-s − 0.143·17-s + 0.00603·18-s + 1.94·19-s − 0.934·21-s − 0.0174·22-s + 0.208·23-s + 0.0282·24-s + 0.00447·26-s + 0.584·27-s − 0.756·28-s + 1.17·29-s − 0.838·31-s + 0.0343·32-s + 1.88·33-s − 0.00163·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 0.0323T + 8T^{2} \) |
| 3 | \( 1 + 6.42T + 27T^{2} \) |
| 7 | \( 1 - 14.0T + 343T^{2} \) |
| 11 | \( 1 + 55.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 161.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 183.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 77.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 315.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 524.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 73.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 132.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 236.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 493.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 806.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 599.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 642.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 883.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 71.2T + 9.12e5T^{2} \) |
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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
−10.16136898227053269485453207302, −8.993120621313102752647960305429, −8.093310150563023079652370058929, −7.25053401144833921067378599445, −5.77586469351680606931170476223, −5.24819368848425502510532262983, −4.60866267959897936291361417436, −3.10772616146280450777366073814, −1.15908943669074590883354976956, 0,
1.15908943669074590883354976956, 3.10772616146280450777366073814, 4.60866267959897936291361417436, 5.24819368848425502510532262983, 5.77586469351680606931170476223, 7.25053401144833921067378599445, 8.093310150563023079652370058929, 8.993120621313102752647960305429, 10.16136898227053269485453207302
