| L(s) = 1 | − 2-s + 2.23·3-s + 4-s + 1.23·5-s − 2.23·6-s − 4.23·7-s − 8-s + 2.00·9-s − 1.23·10-s − 4.23·11-s + 2.23·12-s − 5.47·13-s + 4.23·14-s + 2.76·15-s + 16-s + 2·17-s − 2.00·18-s − 7.70·19-s + 1.23·20-s − 9.47·21-s + 4.23·22-s + 2.23·23-s − 2.23·24-s − 3.47·25-s + 5.47·26-s − 2.23·27-s − 4.23·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.29·3-s + 0.5·4-s + 0.552·5-s − 0.912·6-s − 1.60·7-s − 0.353·8-s + 0.666·9-s − 0.390·10-s − 1.27·11-s + 0.645·12-s − 1.51·13-s + 1.13·14-s + 0.713·15-s + 0.250·16-s + 0.485·17-s − 0.471·18-s − 1.76·19-s + 0.276·20-s − 2.06·21-s + 0.903·22-s + 0.466·23-s − 0.456·24-s − 0.694·25-s + 1.07·26-s − 0.430·27-s − 0.800·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 7.70T + 19T^{2} \) |
| 23 | \( 1 - 2.23T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 7.47T + 61T^{2} \) |
| 67 | \( 1 - 4.70T + 67T^{2} \) |
| 71 | \( 1 - 0.763T + 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
| 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
−9.635105065329912651123167755074, −8.813904836321288961023591255254, −8.024379963505573690350303340547, −7.24640417829595196278935682187, −6.39226919627135280105351293213, −5.34165410433462578805422099238, −3.81029178731685132482729521049, −2.59035053473590255326987416164, −2.41151051338517403888708215935, 0,
2.41151051338517403888708215935, 2.59035053473590255326987416164, 3.81029178731685132482729521049, 5.34165410433462578805422099238, 6.39226919627135280105351293213, 7.24640417829595196278935682187, 8.024379963505573690350303340547, 8.813904836321288961023591255254, 9.635105065329912651123167755074
