| L(s) = 1 | + 2.59·2-s + 4.71·4-s − 1.03·5-s − 1.63·7-s + 7.02·8-s − 2.68·10-s + 1.66·11-s − 4.14·13-s − 4.24·14-s + 8.76·16-s − 2.70·17-s − 19-s − 4.88·20-s + 4.30·22-s − 3.28·23-s − 3.92·25-s − 10.7·26-s − 7.72·28-s − 6.51·29-s − 1.32·31-s + 8.66·32-s − 6.99·34-s + 1.69·35-s + 6.98·37-s − 2.59·38-s − 7.27·40-s − 11.4·41-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 2.35·4-s − 0.463·5-s − 0.619·7-s + 2.48·8-s − 0.849·10-s + 0.501·11-s − 1.15·13-s − 1.13·14-s + 2.19·16-s − 0.655·17-s − 0.229·19-s − 1.09·20-s + 0.917·22-s − 0.684·23-s − 0.785·25-s − 2.10·26-s − 1.45·28-s − 1.21·29-s − 0.238·31-s + 1.53·32-s − 1.19·34-s + 0.287·35-s + 1.14·37-s − 0.420·38-s − 1.15·40-s − 1.79·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 5 | \( 1 + 1.03T + 5T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 13 | \( 1 + 4.14T + 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 + 6.51T + 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 - 6.98T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 8.53T + 43T^{2} \) |
| 53 | \( 1 + 2.69T + 53T^{2} \) |
| 59 | \( 1 - 9.54T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 4.16T + 67T^{2} \) |
| 71 | \( 1 - 8.09T + 71T^{2} \) |
| 73 | \( 1 + 6.28T + 73T^{2} \) |
| 79 | \( 1 - 2.59T + 79T^{2} \) |
| 83 | \( 1 + 8.70T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 5.65T + 97T^{2} \) |
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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
−7.33018266940127561826287825931, −6.51992438756371627504617986248, −6.04971636318339222061366986980, −5.29700746982307049869142279998, −4.51983625265811833921925625183, −3.99860008654592312291084958240, −3.36893404522153260918376730946, −2.51219292859079834013839199977, −1.81206799565312569021985613986, 0,
1.81206799565312569021985613986, 2.51219292859079834013839199977, 3.36893404522153260918376730946, 3.99860008654592312291084958240, 4.51983625265811833921925625183, 5.29700746982307049869142279998, 6.04971636318339222061366986980, 6.51992438756371627504617986248, 7.33018266940127561826287825931
