| L(s) = 1 | + 2.38·3-s − 0.218·5-s − 2.23·7-s + 2.70·9-s − 2.71·11-s + 0.542·13-s − 0.521·15-s + 6.46·17-s − 8.43·19-s − 5.34·21-s + 4.82·23-s − 4.95·25-s − 0.696·27-s − 1.34·29-s − 11.0·31-s − 6.48·33-s + 0.487·35-s − 2.51·37-s + 1.29·39-s + 5.97·41-s − 11.3·43-s − 0.590·45-s − 9.47·47-s − 1.99·49-s + 15.4·51-s + 13.8·53-s + 0.591·55-s + ⋯ |
| L(s) = 1 | + 1.37·3-s − 0.0975·5-s − 0.845·7-s + 0.902·9-s − 0.817·11-s + 0.150·13-s − 0.134·15-s + 1.56·17-s − 1.93·19-s − 1.16·21-s + 1.00·23-s − 0.990·25-s − 0.134·27-s − 0.249·29-s − 1.97·31-s − 1.12·33-s + 0.0824·35-s − 0.414·37-s + 0.207·39-s + 0.933·41-s − 1.73·43-s − 0.0880·45-s − 1.38·47-s − 0.285·49-s + 2.16·51-s + 1.89·53-s + 0.0797·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 2.38T + 3T^{2} \) |
| 5 | \( 1 + 0.218T + 5T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 13 | \( 1 - 0.542T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 + 8.43T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 1.34T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 + 2.51T + 37T^{2} \) |
| 41 | \( 1 - 5.97T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 5.12T + 59T^{2} \) |
| 61 | \( 1 + 1.00T + 61T^{2} \) |
| 67 | \( 1 - 3.27T + 67T^{2} \) |
| 71 | \( 1 - 9.34T + 71T^{2} \) |
| 73 | \( 1 - 2.30T + 73T^{2} \) |
| 79 | \( 1 - 2.50T + 79T^{2} \) |
| 83 | \( 1 - 1.78T + 83T^{2} \) |
| 89 | \( 1 + 8.76T + 89T^{2} \) |
| 97 | \( 1 + 8.59T + 97T^{2} \) |
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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
−8.140350634072325396534674134242, −7.54455688488375162246889461777, −6.79040479180791344496326119470, −5.86760283050668288418295745310, −5.07476879239558612689850569094, −3.80787611723338279253736452549, −3.45321418172884670106053897691, −2.58161326626617781751835491193, −1.75079277155653691387177703913, 0,
1.75079277155653691387177703913, 2.58161326626617781751835491193, 3.45321418172884670106053897691, 3.80787611723338279253736452549, 5.07476879239558612689850569094, 5.86760283050668288418295745310, 6.79040479180791344496326119470, 7.54455688488375162246889461777, 8.140350634072325396534674134242
