L(s) = 1 | + 42.9·2-s + 126.·3-s + 1.33e3·4-s + 2.11e3·5-s + 5.44e3·6-s + 2.40e3·7-s + 3.54e4·8-s − 3.64e3·9-s + 9.10e4·10-s − 2.61e4·11-s + 1.69e5·12-s − 2.85e4·13-s + 1.03e5·14-s + 2.68e5·15-s + 8.40e5·16-s − 6.63e5·17-s − 1.56e5·18-s + 1.33e5·19-s + 2.83e6·20-s + 3.04e5·21-s − 1.12e6·22-s − 1.58e6·23-s + 4.49e6·24-s + 2.53e6·25-s − 1.22e6·26-s − 2.95e6·27-s + 3.20e6·28-s + ⋯ |
L(s) = 1 | + 1.90·2-s + 0.902·3-s + 2.61·4-s + 1.51·5-s + 1.71·6-s + 0.377·7-s + 3.06·8-s − 0.185·9-s + 2.88·10-s − 0.537·11-s + 2.35·12-s − 0.277·13-s + 0.718·14-s + 1.36·15-s + 3.20·16-s − 1.92·17-s − 0.352·18-s + 0.235·19-s + 3.95·20-s + 0.341·21-s − 1.02·22-s − 1.18·23-s + 2.76·24-s + 1.29·25-s − 0.527·26-s − 1.06·27-s + 0.986·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(11.35611006\) |
\(L(\frac12)\) |
\(\approx\) |
\(11.35611006\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 + 2.85e4T \) |
good | 2 | \( 1 - 42.9T + 512T^{2} \) |
| 3 | \( 1 - 126.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.11e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 2.61e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 6.63e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.33e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.58e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.21e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.17e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.05e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.61e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.59e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.57e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.74e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.30e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.00e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 5.80e6T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.08e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.12e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.82e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.64e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.18e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.62e9T + 7.60e17T^{2} \) |
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
−12.84661038748370565271525588870, −11.44764902881905359397436917534, −10.38006147073947178410532294418, −8.940194431454610877385921383297, −7.34725201153403167374916293079, −6.06066486804745436748826799654, −5.25027980208854761276566518699, −3.91228243931704024601108672471, −2.29944392306766621666166347430, −2.18335528961099792398081017143,
2.18335528961099792398081017143, 2.29944392306766621666166347430, 3.91228243931704024601108672471, 5.25027980208854761276566518699, 6.06066486804745436748826799654, 7.34725201153403167374916293079, 8.940194431454610877385921383297, 10.38006147073947178410532294418, 11.44764902881905359397436917534, 12.84661038748370565271525588870