| L(s) = 1 | − 0.887·2-s − 2.66·3-s − 1.21·4-s − 0.744·5-s + 2.36·6-s − 0.702·7-s + 2.85·8-s + 4.08·9-s + 0.661·10-s + 1.89·11-s + 3.22·12-s + 5.87·13-s + 0.623·14-s + 1.98·15-s − 0.105·16-s − 6.29·17-s − 3.62·18-s − 0.340·19-s + 0.903·20-s + 1.86·21-s − 1.67·22-s + 4.82·23-s − 7.58·24-s − 4.44·25-s − 5.21·26-s − 2.88·27-s + 0.851·28-s + ⋯ |
| L(s) = 1 | − 0.627·2-s − 1.53·3-s − 0.606·4-s − 0.333·5-s + 0.964·6-s − 0.265·7-s + 1.00·8-s + 1.36·9-s + 0.209·10-s + 0.569·11-s + 0.931·12-s + 1.63·13-s + 0.166·14-s + 0.511·15-s − 0.0262·16-s − 1.52·17-s − 0.854·18-s − 0.0782·19-s + 0.201·20-s + 0.407·21-s − 0.357·22-s + 1.00·23-s − 1.54·24-s − 0.889·25-s − 1.02·26-s − 0.555·27-s + 0.160·28-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 | \( 1 + 0.887T + 2T^{2} \) |
| 3 | \( 1 + 2.66T + 3T^{2} \) |
| 5 | \( 1 + 0.744T + 5T^{2} \) |
| 7 | \( 1 + 0.702T + 7T^{2} \) |
| 11 | \( 1 - 1.89T + 11T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 17 | \( 1 + 6.29T + 17T^{2} \) |
| 19 | \( 1 + 0.340T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 1.90T + 31T^{2} \) |
| 37 | \( 1 + 3.11T + 37T^{2} \) |
| 41 | \( 1 + 3.21T + 41T^{2} \) |
| 43 | \( 1 - 7.84T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 6.85T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 + 5.99T + 67T^{2} \) |
| 71 | \( 1 + 9.17T + 71T^{2} \) |
| 73 | \( 1 - 7.10T + 73T^{2} \) |
| 79 | \( 1 + 3.80T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + 1.94T + 89T^{2} \) |
| 97 | \( 1 - 9.71T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.40652217263511581642809155954, −9.210583884996555091021006980416, −8.710523126662996467335193601226, −7.44450278343688003655445836176, −6.50124383727195818757493297581, −5.73553170568928121098427478037, −4.59854010130423032596789882873, −3.81738243939927589968094830191, −1.34945254892981540722578252430, 0,
1.34945254892981540722578252430, 3.81738243939927589968094830191, 4.59854010130423032596789882873, 5.73553170568928121098427478037, 6.50124383727195818757493297581, 7.44450278343688003655445836176, 8.710523126662996467335193601226, 9.210583884996555091021006980416, 10.40652217263511581642809155954
