L(s) = 1 | − 2.70·2-s + 5.33·4-s − 2.87·5-s + 7-s − 9.04·8-s + 7.79·10-s + 1.50·11-s + 5.43·13-s − 2.70·14-s + 13.8·16-s − 1.68·17-s − 3.30·19-s − 15.3·20-s − 4.06·22-s + 5.07·23-s + 3.28·25-s − 14.7·26-s + 5.33·28-s + 8.28·29-s − 9.29·31-s − 19.3·32-s + 4.56·34-s − 2.87·35-s − 7.22·37-s + 8.94·38-s + 26.0·40-s + 2.03·41-s + ⋯ |
L(s) = 1 | − 1.91·2-s + 2.66·4-s − 1.28·5-s + 0.377·7-s − 3.19·8-s + 2.46·10-s + 0.452·11-s + 1.50·13-s − 0.723·14-s + 3.45·16-s − 0.408·17-s − 0.757·19-s − 3.43·20-s − 0.866·22-s + 1.05·23-s + 0.657·25-s − 2.88·26-s + 1.00·28-s + 1.53·29-s − 1.66·31-s − 3.41·32-s + 0.782·34-s − 0.486·35-s − 1.18·37-s + 1.45·38-s + 4.11·40-s + 0.317·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 11 | \( 1 - 1.50T + 11T^{2} \) |
| 13 | \( 1 - 5.43T + 13T^{2} \) |
| 17 | \( 1 + 1.68T + 17T^{2} \) |
| 19 | \( 1 + 3.30T + 19T^{2} \) |
| 23 | \( 1 - 5.07T + 23T^{2} \) |
| 29 | \( 1 - 8.28T + 29T^{2} \) |
| 31 | \( 1 + 9.29T + 31T^{2} \) |
| 37 | \( 1 + 7.22T + 37T^{2} \) |
| 41 | \( 1 - 2.03T + 41T^{2} \) |
| 43 | \( 1 - 0.859T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 8.32T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 9.99T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 + 5.53T + 71T^{2} \) |
| 73 | \( 1 - 6.56T + 73T^{2} \) |
| 79 | \( 1 + 9.27T + 79T^{2} \) |
| 83 | \( 1 + 7.17T + 83T^{2} \) |
| 89 | \( 1 + 9.76T + 89T^{2} \) |
| 97 | \( 1 + 9.08T + 97T^{2} \) |
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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
−7.63227367376549006477614539478, −7.11589163697469304633968158694, −6.50333343299231071566135160526, −5.77964970785956133227289661155, −4.51150545462456089130948811328, −3.66292437634884631888547891532, −2.93663163621314274017159023406, −1.76761288060562782071768442327, −1.03816421937151758153099926746, 0,
1.03816421937151758153099926746, 1.76761288060562782071768442327, 2.93663163621314274017159023406, 3.66292437634884631888547891532, 4.51150545462456089130948811328, 5.77964970785956133227289661155, 6.50333343299231071566135160526, 7.11589163697469304633968158694, 7.63227367376549006477614539478