| L(s) = 1 | + 0.589·3-s − 1.64·5-s + 1.73·7-s − 2.65·9-s + 4.09·11-s − 2.54·13-s − 0.969·15-s − 0.508·17-s − 0.571·19-s + 1.02·21-s − 4.07·23-s − 2.29·25-s − 3.33·27-s + 3.73·29-s + 2.84·31-s + 2.41·33-s − 2.85·35-s + 5.86·37-s − 1.50·39-s + 5.85·41-s − 11.2·43-s + 4.36·45-s + 1.87·47-s − 3.99·49-s − 0.299·51-s − 12.1·53-s − 6.72·55-s + ⋯ |
| L(s) = 1 | + 0.340·3-s − 0.735·5-s + 0.655·7-s − 0.884·9-s + 1.23·11-s − 0.705·13-s − 0.250·15-s − 0.123·17-s − 0.131·19-s + 0.223·21-s − 0.850·23-s − 0.459·25-s − 0.641·27-s + 0.693·29-s + 0.510·31-s + 0.419·33-s − 0.481·35-s + 0.963·37-s − 0.240·39-s + 0.915·41-s − 1.71·43-s + 0.650·45-s + 0.272·47-s − 0.570·49-s − 0.0420·51-s − 1.67·53-s − 0.906·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 0.589T + 3T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 - 4.09T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 + 0.508T + 17T^{2} \) |
| 19 | \( 1 + 0.571T + 19T^{2} \) |
| 23 | \( 1 + 4.07T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 - 2.84T + 31T^{2} \) |
| 37 | \( 1 - 5.86T + 37T^{2} \) |
| 41 | \( 1 - 5.85T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 1.87T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 0.487T + 61T^{2} \) |
| 67 | \( 1 - 6.83T + 67T^{2} \) |
| 71 | \( 1 + 1.66T + 71T^{2} \) |
| 73 | \( 1 - 5.33T + 73T^{2} \) |
| 79 | \( 1 + 8.87T + 79T^{2} \) |
| 83 | \( 1 + 3.35T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.071380066451067326934533454392, −7.60757031863290755662353248133, −6.57590417638819961332814883985, −5.98054873901449646277007786978, −4.87771931003989223774842687764, −4.26504122033793036291163205022, −3.44768052930218652177526132037, −2.52833493871770255678974977118, −1.46943544490769101040674591048, 0,
1.46943544490769101040674591048, 2.52833493871770255678974977118, 3.44768052930218652177526132037, 4.26504122033793036291163205022, 4.87771931003989223774842687764, 5.98054873901449646277007786978, 6.57590417638819961332814883985, 7.60757031863290755662353248133, 8.071380066451067326934533454392
