| L(s) = 1 | + 0.477·2-s + 0.323·3-s − 1.77·4-s − 5-s + 0.154·6-s + 2.68·7-s − 1.80·8-s − 2.89·9-s − 0.477·10-s − 0.572·12-s − 4.66·13-s + 1.28·14-s − 0.323·15-s + 2.68·16-s − 4.62·17-s − 1.38·18-s − 4.34·19-s + 1.77·20-s + 0.867·21-s + 2.77·23-s − 0.581·24-s + 25-s − 2.22·26-s − 1.90·27-s − 4.75·28-s − 3.01·29-s − 0.154·30-s + ⋯ |
| L(s) = 1 | + 0.337·2-s + 0.186·3-s − 0.886·4-s − 0.447·5-s + 0.0629·6-s + 1.01·7-s − 0.636·8-s − 0.965·9-s − 0.150·10-s − 0.165·12-s − 1.29·13-s + 0.342·14-s − 0.0834·15-s + 0.671·16-s − 1.12·17-s − 0.325·18-s − 0.995·19-s + 0.396·20-s + 0.189·21-s + 0.578·23-s − 0.118·24-s + 0.200·25-s − 0.436·26-s − 0.366·27-s − 0.899·28-s − 0.559·29-s − 0.0281·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.477T + 2T^{2} \) |
| 3 | \( 1 - 0.323T + 3T^{2} \) |
| 7 | \( 1 - 2.68T + 7T^{2} \) |
| 13 | \( 1 + 4.66T + 13T^{2} \) |
| 17 | \( 1 + 4.62T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 - 2.77T + 23T^{2} \) |
| 29 | \( 1 + 3.01T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 2.21T + 41T^{2} \) |
| 43 | \( 1 + 7.06T + 43T^{2} \) |
| 47 | \( 1 - 4.36T + 47T^{2} \) |
| 53 | \( 1 + 6.33T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 - 7.31T + 67T^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 - 1.02T + 73T^{2} \) |
| 79 | \( 1 - 3.50T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 2.76T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
| 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
−10.25769551484400349152082152564, −9.057686604933394976872546813950, −8.559602694378998320968739952163, −7.77042925084168389491816674765, −6.58072521883113617915473801395, −5.19052348752579281999339725195, −4.74965239119804516306382767148, −3.60592616505334352342799589288, −2.26687467016727898820150766433, 0,
2.26687467016727898820150766433, 3.60592616505334352342799589288, 4.74965239119804516306382767148, 5.19052348752579281999339725195, 6.58072521883113617915473801395, 7.77042925084168389491816674765, 8.559602694378998320968739952163, 9.057686604933394976872546813950, 10.25769551484400349152082152564
