| L(s) = 1 | − 0.232·2-s − 1.45·3-s − 1.94·4-s + 0.338·6-s + 4.96·7-s + 0.919·8-s − 0.892·9-s − 2.05·11-s + 2.82·12-s − 1.61·13-s − 1.15·14-s + 3.67·16-s + 7.51·17-s + 0.207·18-s − 5.06·19-s − 7.20·21-s + 0.478·22-s − 1.87·23-s − 1.33·24-s + 0.377·26-s + 5.65·27-s − 9.65·28-s + 2.47·29-s − 10.3·31-s − 2.69·32-s + 2.98·33-s − 1.75·34-s + ⋯ |
| L(s) = 1 | − 0.164·2-s − 0.838·3-s − 0.972·4-s + 0.138·6-s + 1.87·7-s + 0.324·8-s − 0.297·9-s − 0.619·11-s + 0.815·12-s − 0.449·13-s − 0.309·14-s + 0.919·16-s + 1.82·17-s + 0.0489·18-s − 1.16·19-s − 1.57·21-s + 0.102·22-s − 0.391·23-s − 0.272·24-s + 0.0739·26-s + 1.08·27-s − 1.82·28-s + 0.460·29-s − 1.86·31-s − 0.476·32-s + 0.519·33-s − 0.300·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 0.232T + 2T^{2} \) |
| 3 | \( 1 + 1.45T + 3T^{2} \) |
| 7 | \( 1 - 4.96T + 7T^{2} \) |
| 11 | \( 1 + 2.05T + 11T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 - 7.51T + 17T^{2} \) |
| 19 | \( 1 + 5.06T + 19T^{2} \) |
| 23 | \( 1 + 1.87T + 23T^{2} \) |
| 29 | \( 1 - 2.47T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 + 6.75T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 0.547T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 2.84T + 71T^{2} \) |
| 73 | \( 1 - 1.15T + 73T^{2} \) |
| 79 | \( 1 + 7.92T + 79T^{2} \) |
| 83 | \( 1 + 1.24T + 83T^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 97 | \( 1 + 2.98T + 97T^{2} \) |
| 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
−7.84567588200517301951229941893, −7.25609473196935414263886136099, −5.96541668146009113137239584481, −5.40103408553825845608378133417, −4.98600754294511998791586994337, −4.36287617274402495442218792913, −3.36327416451737724687670866106, −2.06537364540763368643471131185, −1.13077894622723713579856501283, 0,
1.13077894622723713579856501283, 2.06537364540763368643471131185, 3.36327416451737724687670866106, 4.36287617274402495442218792913, 4.98600754294511998791586994337, 5.40103408553825845608378133417, 5.96541668146009113137239584481, 7.25609473196935414263886136099, 7.84567588200517301951229941893
