L(s) = 1 | − 2-s + 3-s + 4-s − 3.78·5-s − 6-s + 0.659·7-s − 8-s + 9-s + 3.78·10-s − 2.85·11-s + 12-s + 1.83·13-s − 0.659·14-s − 3.78·15-s + 16-s + 17-s − 18-s + 3.09·19-s − 3.78·20-s + 0.659·21-s + 2.85·22-s − 4.72·23-s − 24-s + 9.35·25-s − 1.83·26-s + 27-s + 0.659·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.69·5-s − 0.408·6-s + 0.249·7-s − 0.353·8-s + 0.333·9-s + 1.19·10-s − 0.861·11-s + 0.288·12-s + 0.508·13-s − 0.176·14-s − 0.978·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.710·19-s − 0.847·20-s + 0.143·21-s + 0.609·22-s − 0.985·23-s − 0.204·24-s + 1.87·25-s − 0.359·26-s + 0.192·27-s + 0.124·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 + 3.78T + 5T^{2} \) |
| 7 | \( 1 - 0.659T + 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 19 | \( 1 - 3.09T + 19T^{2} \) |
| 23 | \( 1 + 4.72T + 23T^{2} \) |
| 29 | \( 1 + 3.49T + 29T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 - 3.81T + 37T^{2} \) |
| 41 | \( 1 - 1.56T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 + 2.29T + 47T^{2} \) |
| 53 | \( 1 - 5.66T + 53T^{2} \) |
| 61 | \( 1 - 9.74T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 6.95T + 73T^{2} \) |
| 79 | \( 1 - 8.69T + 79T^{2} \) |
| 83 | \( 1 + 6.90T + 83T^{2} \) |
| 89 | \( 1 + 0.203T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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
−7.913680395419732178705865924499, −7.41975062841528431748478602270, −6.61902490800483361543488999442, −5.59038891174480346404503123574, −4.66931842820872268823805230884, −3.84163845241693536640970497405, −3.25791559618794977229702148869, −2.35714034266682429016086595919, −1.12789668312220999079435788608, 0,
1.12789668312220999079435788608, 2.35714034266682429016086595919, 3.25791559618794977229702148869, 3.84163845241693536640970497405, 4.66931842820872268823805230884, 5.59038891174480346404503123574, 6.61902490800483361543488999442, 7.41975062841528431748478602270, 7.913680395419732178705865924499