| L(s) = 1 | − 0.723·2-s + 3.01·3-s − 1.47·4-s − 2.02·5-s − 2.18·6-s − 2.86·7-s + 2.51·8-s + 6.10·9-s + 1.46·10-s − 5.18·11-s − 4.45·12-s − 4.90·13-s + 2.07·14-s − 6.12·15-s + 1.12·16-s + 7.18·17-s − 4.41·18-s − 2.17·19-s + 2.99·20-s − 8.65·21-s + 3.75·22-s − 7.89·23-s + 7.59·24-s − 0.879·25-s + 3.55·26-s + 9.36·27-s + 4.23·28-s + ⋯ |
| L(s) = 1 | − 0.511·2-s + 1.74·3-s − 0.737·4-s − 0.907·5-s − 0.891·6-s − 1.08·7-s + 0.889·8-s + 2.03·9-s + 0.464·10-s − 1.56·11-s − 1.28·12-s − 1.36·13-s + 0.554·14-s − 1.58·15-s + 0.282·16-s + 1.74·17-s − 1.04·18-s − 0.498·19-s + 0.669·20-s − 1.88·21-s + 0.800·22-s − 1.64·23-s + 1.54·24-s − 0.175·25-s + 0.696·26-s + 1.80·27-s + 0.799·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 619 | \( 1 + T \) |
| good | 2 | \( 1 + 0.723T + 2T^{2} \) |
| 3 | \( 1 - 3.01T + 3T^{2} \) |
| 5 | \( 1 + 2.02T + 5T^{2} \) |
| 7 | \( 1 + 2.86T + 7T^{2} \) |
| 11 | \( 1 + 5.18T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 - 7.18T + 17T^{2} \) |
| 19 | \( 1 + 2.17T + 19T^{2} \) |
| 23 | \( 1 + 7.89T + 23T^{2} \) |
| 29 | \( 1 + 7.73T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 + 9.16T + 37T^{2} \) |
| 41 | \( 1 - 3.94T + 41T^{2} \) |
| 43 | \( 1 - 7.18T + 43T^{2} \) |
| 47 | \( 1 - 3.57T + 47T^{2} \) |
| 53 | \( 1 - 4.73T + 53T^{2} \) |
| 59 | \( 1 + 8.30T + 59T^{2} \) |
| 61 | \( 1 - 1.43T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 6.02T + 71T^{2} \) |
| 73 | \( 1 + 2.63T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 9.64T + 83T^{2} \) |
| 89 | \( 1 + 6.74T + 89T^{2} \) |
| 97 | \( 1 - 2.43T + 97T^{2} \) |
| 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
−9.879762115889498932428548214446, −9.400320872063317087637269072750, −8.259719901110090097354015267140, −7.78422354744040172147949610288, −7.36245077150887160693635788832, −5.43135120865070755287628455710, −4.11962901152161918900502254858, −3.40321925465945160531643690324, −2.33067517151161547199285324004, 0,
2.33067517151161547199285324004, 3.40321925465945160531643690324, 4.11962901152161918900502254858, 5.43135120865070755287628455710, 7.36245077150887160693635788832, 7.78422354744040172147949610288, 8.259719901110090097354015267140, 9.400320872063317087637269072750, 9.879762115889498932428548214446
