| L(s) = 1 | − 0.587·2-s − 3-s − 1.65·4-s − 5-s + 0.587·6-s − 1.98·7-s + 2.14·8-s + 9-s + 0.587·10-s + 1.32·11-s + 1.65·12-s − 4.37·13-s + 1.16·14-s + 15-s + 2.04·16-s + 6.01·17-s − 0.587·18-s − 8.00·19-s + 1.65·20-s + 1.98·21-s − 0.778·22-s − 2.14·24-s + 25-s + 2.57·26-s − 27-s + 3.27·28-s − 1.61·29-s + ⋯ |
| L(s) = 1 | − 0.415·2-s − 0.577·3-s − 0.827·4-s − 0.447·5-s + 0.239·6-s − 0.748·7-s + 0.758·8-s + 0.333·9-s + 0.185·10-s + 0.399·11-s + 0.477·12-s − 1.21·13-s + 0.310·14-s + 0.258·15-s + 0.512·16-s + 1.45·17-s − 0.138·18-s − 1.83·19-s + 0.370·20-s + 0.432·21-s − 0.166·22-s − 0.438·24-s + 0.200·25-s + 0.504·26-s − 0.192·27-s + 0.619·28-s − 0.300·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3492732146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3492732146\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 0.587T + 2T^{2} \) |
| 7 | \( 1 + 1.98T + 7T^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 - 6.01T + 17T^{2} \) |
| 19 | \( 1 + 8.00T + 19T^{2} \) |
| 29 | \( 1 + 1.61T + 29T^{2} \) |
| 31 | \( 1 - 1.47T + 31T^{2} \) |
| 37 | \( 1 + 0.666T + 37T^{2} \) |
| 41 | \( 1 - 3.57T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 9.62T + 47T^{2} \) |
| 53 | \( 1 - 2.20T + 53T^{2} \) |
| 59 | \( 1 - 7.22T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 6.96T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 5.21T + 89T^{2} \) |
| 97 | \( 1 + 5.95T + 97T^{2} \) |
| 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
−7.84726061530339203059168258135, −7.25935404947269705258941756423, −6.47637348342092038268559852178, −5.77516073665528379962446870241, −4.97522573232345073548822989202, −4.32035273422165651988938751761, −3.70503714542788942094661612519, −2.71412852082405779699678478256, −1.45510125874186804402423328398, −0.33972012473053632970537865919,
0.33972012473053632970537865919, 1.45510125874186804402423328398, 2.71412852082405779699678478256, 3.70503714542788942094661612519, 4.32035273422165651988938751761, 4.97522573232345073548822989202, 5.77516073665528379962446870241, 6.47637348342092038268559852178, 7.25935404947269705258941756423, 7.84726061530339203059168258135
