| L(s) = 1 | + 0.288·2-s + 3-s − 1.91·4-s + 0.288·6-s + 3.66·7-s − 1.13·8-s + 9-s − 1.91·12-s − 4.82·13-s + 1.05·14-s + 3.50·16-s − 3.84·17-s + 0.288·18-s − 6.96·19-s + 3.66·21-s + 1.20·23-s − 1.13·24-s − 1.39·26-s + 27-s − 7.02·28-s + 4.94·29-s + 7.36·31-s + 3.27·32-s − 1.10·34-s − 1.91·36-s + 2.38·37-s − 2.01·38-s + ⋯ |
| L(s) = 1 | + 0.204·2-s + 0.577·3-s − 0.958·4-s + 0.117·6-s + 1.38·7-s − 0.399·8-s + 0.333·9-s − 0.553·12-s − 1.33·13-s + 0.283·14-s + 0.876·16-s − 0.931·17-s + 0.0680·18-s − 1.59·19-s + 0.800·21-s + 0.251·23-s − 0.230·24-s − 0.273·26-s + 0.192·27-s − 1.32·28-s + 0.918·29-s + 1.32·31-s + 0.578·32-s − 0.190·34-s − 0.319·36-s + 0.392·37-s − 0.326·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.288T + 2T^{2} \) |
| 7 | \( 1 - 3.66T + 7T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 3.84T + 17T^{2} \) |
| 19 | \( 1 + 6.96T + 19T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 - 4.94T + 29T^{2} \) |
| 31 | \( 1 - 7.36T + 31T^{2} \) |
| 37 | \( 1 - 2.38T + 37T^{2} \) |
| 41 | \( 1 - 4.30T + 41T^{2} \) |
| 43 | \( 1 + 0.772T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 + 6.10T + 67T^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 + 0.191T + 73T^{2} \) |
| 79 | \( 1 + 3.15T + 79T^{2} \) |
| 83 | \( 1 + 1.53T + 83T^{2} \) |
| 89 | \( 1 + 3.04T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
| 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
−7.67443719997100647914722491799, −6.73205950990204403702426945850, −5.96690780110197999702368168100, −4.94146600380001712187400905109, −4.48778326658143386444234314317, −4.28123557002971263282543857183, −2.93482980084969931653380953839, −2.31035773309968291448736579489, −1.31946457096685034629791244501, 0,
1.31946457096685034629791244501, 2.31035773309968291448736579489, 2.93482980084969931653380953839, 4.28123557002971263282543857183, 4.48778326658143386444234314317, 4.94146600380001712187400905109, 5.96690780110197999702368168100, 6.73205950990204403702426945850, 7.67443719997100647914722491799
