| L(s) = 1 | − 1.23·2-s − 0.464·4-s + 0.602·5-s + 1.86·7-s + 3.05·8-s − 0.746·10-s − 3.17·11-s + 1.00·13-s − 2.30·14-s − 2.85·16-s + 1.11·17-s − 0.230·19-s − 0.279·20-s + 3.93·22-s − 1.82·23-s − 4.63·25-s − 1.24·26-s − 0.863·28-s − 0.299·29-s + 9.32·31-s − 2.56·32-s − 1.38·34-s + 1.12·35-s + 0.291·37-s + 0.285·38-s + 1.83·40-s − 6.20·41-s + ⋯ |
| L(s) = 1 | − 0.876·2-s − 0.232·4-s + 0.269·5-s + 0.703·7-s + 1.07·8-s − 0.236·10-s − 0.956·11-s + 0.279·13-s − 0.616·14-s − 0.713·16-s + 0.271·17-s − 0.0528·19-s − 0.0625·20-s + 0.838·22-s − 0.381·23-s − 0.927·25-s − 0.244·26-s − 0.163·28-s − 0.0555·29-s + 1.67·31-s − 0.454·32-s − 0.237·34-s + 0.189·35-s + 0.0479·37-s + 0.0463·38-s + 0.290·40-s − 0.969·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 + 1.23T + 2T^{2} \) |
| 5 | \( 1 - 0.602T + 5T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 + 3.17T + 11T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 + 0.230T + 19T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 29 | \( 1 + 0.299T + 29T^{2} \) |
| 31 | \( 1 - 9.32T + 31T^{2} \) |
| 37 | \( 1 - 0.291T + 37T^{2} \) |
| 41 | \( 1 + 6.20T + 41T^{2} \) |
| 43 | \( 1 + 9.62T + 43T^{2} \) |
| 47 | \( 1 + 4.16T + 47T^{2} \) |
| 53 | \( 1 + 0.477T + 53T^{2} \) |
| 59 | \( 1 - 0.954T + 59T^{2} \) |
| 61 | \( 1 - 2.26T + 61T^{2} \) |
| 67 | \( 1 + 1.26T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 8.76T + 73T^{2} \) |
| 79 | \( 1 - 5.73T + 79T^{2} \) |
| 83 | \( 1 + 8.33T + 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−8.290000967635285517341437455973, −7.67359378220637191627277611480, −6.79585389462572481996509034154, −5.82516410112702846480885828005, −5.01801774992621418103350281169, −4.44781039271858428035951502171, −3.33789861307937463118298827045, −2.15115515410324035069880288931, −1.31054171399185823572190204276, 0,
1.31054171399185823572190204276, 2.15115515410324035069880288931, 3.33789861307937463118298827045, 4.44781039271858428035951502171, 5.01801774992621418103350281169, 5.82516410112702846480885828005, 6.79585389462572481996509034154, 7.67359378220637191627277611480, 8.290000967635285517341437455973
