| L(s) = 1 | + 2.79·2-s − 1.06·3-s + 5.80·4-s − 5-s − 2.98·6-s − 7-s + 10.6·8-s − 1.86·9-s − 2.79·10-s + 0.504·11-s − 6.19·12-s + 2.67·13-s − 2.79·14-s + 1.06·15-s + 18.0·16-s + 0.222·17-s − 5.19·18-s + 3.37·19-s − 5.80·20-s + 1.06·21-s + 1.40·22-s − 8.15·23-s − 11.3·24-s + 25-s + 7.45·26-s + 5.18·27-s − 5.80·28-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 0.616·3-s + 2.90·4-s − 0.447·5-s − 1.21·6-s − 0.377·7-s + 3.75·8-s − 0.620·9-s − 0.883·10-s + 0.152·11-s − 1.78·12-s + 0.740·13-s − 0.746·14-s + 0.275·15-s + 4.51·16-s + 0.0539·17-s − 1.22·18-s + 0.773·19-s − 1.29·20-s + 0.232·21-s + 0.300·22-s − 1.69·23-s − 2.31·24-s + 0.200·25-s + 1.46·26-s + 0.998·27-s − 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.045475786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.045475786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 3 | \( 1 + 1.06T + 3T^{2} \) |
| 11 | \( 1 - 0.504T + 11T^{2} \) |
| 13 | \( 1 - 2.67T + 13T^{2} \) |
| 17 | \( 1 - 0.222T + 17T^{2} \) |
| 19 | \( 1 - 3.37T + 19T^{2} \) |
| 23 | \( 1 + 8.15T + 23T^{2} \) |
| 29 | \( 1 + 7.54T + 29T^{2} \) |
| 31 | \( 1 - 9.86T + 31T^{2} \) |
| 37 | \( 1 + 5.73T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 2.74T + 43T^{2} \) |
| 47 | \( 1 - 8.69T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 5.57T + 61T^{2} \) |
| 67 | \( 1 - 8.95T + 67T^{2} \) |
| 71 | \( 1 + 0.120T + 71T^{2} \) |
| 73 | \( 1 - 6.77T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 4.74T + 83T^{2} \) |
| 89 | \( 1 + 0.176T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.60471336756955240125982405512, −6.71754787458915898741256086329, −6.18448947864870020687679645080, −5.67519788070530709897174616538, −5.16009877915293059445621422975, −4.11285925737183191949811788600, −3.83888487050058047916784133061, −2.94843738447287669768292845146, −2.20917721885715500841359517566, −0.938844504363102469891245383241,
0.938844504363102469891245383241, 2.20917721885715500841359517566, 2.94843738447287669768292845146, 3.83888487050058047916784133061, 4.11285925737183191949811788600, 5.16009877915293059445621422975, 5.67519788070530709897174616538, 6.18448947864870020687679645080, 6.71754787458915898741256086329, 7.60471336756955240125982405512
