L(s) = 1 | − 2.02·2-s + 3-s + 2.11·4-s + 4.33·5-s − 2.02·6-s − 3.04·7-s − 0.237·8-s + 9-s − 8.80·10-s − 2.48·11-s + 2.11·12-s − 2.83·13-s + 6.17·14-s + 4.33·15-s − 3.75·16-s + 17-s − 2.02·18-s − 1.71·19-s + 9.18·20-s − 3.04·21-s + 5.03·22-s − 2.49·23-s − 0.237·24-s + 13.8·25-s + 5.75·26-s + 27-s − 6.44·28-s + ⋯ |
L(s) = 1 | − 1.43·2-s + 0.577·3-s + 1.05·4-s + 1.93·5-s − 0.828·6-s − 1.14·7-s − 0.0838·8-s + 0.333·9-s − 2.78·10-s − 0.748·11-s + 0.611·12-s − 0.786·13-s + 1.64·14-s + 1.11·15-s − 0.938·16-s + 0.242·17-s − 0.478·18-s − 0.393·19-s + 2.05·20-s − 0.663·21-s + 1.07·22-s − 0.520·23-s − 0.0484·24-s + 2.76·25-s + 1.12·26-s + 0.192·27-s − 1.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.309099363\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.309099363\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 2.02T + 2T^{2} \) |
| 5 | \( 1 - 4.33T + 5T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 + 2.83T + 13T^{2} \) |
| 19 | \( 1 + 1.71T + 19T^{2} \) |
| 23 | \( 1 + 2.49T + 23T^{2} \) |
| 29 | \( 1 + 2.97T + 29T^{2} \) |
| 31 | \( 1 + 0.206T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 1.56T + 47T^{2} \) |
| 53 | \( 1 + 3.33T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 3.95T + 67T^{2} \) |
| 71 | \( 1 - 7.14T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 3.69T + 79T^{2} \) |
| 83 | \( 1 + 1.81T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 9.74T + 97T^{2} \) |
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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
−7.82216586781283995256999504226, −7.42070602515509398849623212750, −6.47463559812969083715751571283, −6.09203833570065940021130637004, −5.24064168093649852387233433962, −4.29153147684942553874945351818, −2.90718901876812365308503086639, −2.45873948692117191384235441176, −1.79090819298121824511912246311, −0.67091892595456495645680995935,
0.67091892595456495645680995935, 1.79090819298121824511912246311, 2.45873948692117191384235441176, 2.90718901876812365308503086639, 4.29153147684942553874945351818, 5.24064168093649852387233433962, 6.09203833570065940021130637004, 6.47463559812969083715751571283, 7.42070602515509398849623212750, 7.82216586781283995256999504226