| L(s) = 1 | + 2-s + 4-s + 2.35·5-s − 1.22·7-s + 8-s + 2.35·10-s − 5.87·11-s − 3.29·13-s − 1.22·14-s + 16-s + 1.69·17-s − 0.613·19-s + 2.35·20-s − 5.87·22-s − 5.52·23-s + 0.523·25-s − 3.29·26-s − 1.22·28-s − 6.27·29-s − 1.38·31-s + 32-s + 1.69·34-s − 2.89·35-s + 2.93·37-s − 0.613·38-s + 2.35·40-s + 2.48·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.05·5-s − 0.464·7-s + 0.353·8-s + 0.743·10-s − 1.77·11-s − 0.913·13-s − 0.328·14-s + 0.250·16-s + 0.411·17-s − 0.140·19-s + 0.525·20-s − 1.25·22-s − 1.15·23-s + 0.104·25-s − 0.645·26-s − 0.232·28-s − 1.16·29-s − 0.248·31-s + 0.176·32-s + 0.290·34-s − 0.488·35-s + 0.481·37-s − 0.0995·38-s + 0.371·40-s + 0.388·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 - 2.35T + 5T^{2} \) |
| 7 | \( 1 + 1.22T + 7T^{2} \) |
| 11 | \( 1 + 5.87T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 17 | \( 1 - 1.69T + 17T^{2} \) |
| 19 | \( 1 + 0.613T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 29 | \( 1 + 6.27T + 29T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 - 2.93T + 37T^{2} \) |
| 41 | \( 1 - 2.48T + 41T^{2} \) |
| 43 | \( 1 + 3.47T + 43T^{2} \) |
| 47 | \( 1 - 0.843T + 47T^{2} \) |
| 53 | \( 1 - 3.77T + 53T^{2} \) |
| 59 | \( 1 - 4.43T + 59T^{2} \) |
| 61 | \( 1 + 7.93T + 61T^{2} \) |
| 67 | \( 1 + 8.35T + 67T^{2} \) |
| 71 | \( 1 + 7.83T + 71T^{2} \) |
| 73 | \( 1 + 6.71T + 73T^{2} \) |
| 79 | \( 1 - 0.124T + 79T^{2} \) |
| 83 | \( 1 - 0.936T + 83T^{2} \) |
| 89 | \( 1 - 9.09T + 89T^{2} \) |
| 97 | \( 1 - 1.28T + 97T^{2} \) |
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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
−7.74097875486068646881743005539, −7.46057003569035182606238991286, −6.32783990413348826683985525312, −5.73945024892319749640850179828, −5.24689427694212538791892563048, −4.40459894800188114180195460974, −3.28182365885607572341236543046, −2.50046153014486675826209355010, −1.86095785676571554135156501400, 0,
1.86095785676571554135156501400, 2.50046153014486675826209355010, 3.28182365885607572341236543046, 4.40459894800188114180195460974, 5.24689427694212538791892563048, 5.73945024892319749640850179828, 6.32783990413348826683985525312, 7.46057003569035182606238991286, 7.74097875486068646881743005539
