| L(s) = 1 | − 1.61·2-s + 0.618·4-s + 5-s + 2.23·8-s − 1.61·10-s − 11-s + 5.47·13-s − 4.85·16-s + 0.763·17-s − 6.70·19-s + 0.618·20-s + 1.61·22-s + 7.70·23-s − 4·25-s − 8.85·26-s − 5·29-s + 0.763·31-s + 3.38·32-s − 1.23·34-s − 7·37-s + 10.8·38-s + 2.23·40-s + 6.47·41-s − 7.70·43-s − 0.618·44-s − 12.4·46-s − 4.23·47-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s + 0.447·5-s + 0.790·8-s − 0.511·10-s − 0.301·11-s + 1.51·13-s − 1.21·16-s + 0.185·17-s − 1.53·19-s + 0.138·20-s + 0.344·22-s + 1.60·23-s − 0.800·25-s − 1.73·26-s − 0.928·29-s + 0.137·31-s + 0.597·32-s − 0.211·34-s − 1.15·37-s + 1.76·38-s + 0.353·40-s + 1.01·41-s − 1.17·43-s − 0.0931·44-s − 1.83·46-s − 0.617·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 0.763T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 5.52T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 - 3.70T + 97T^{2} \) |
| 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
−8.101701226178540212510230196486, −7.38605823321025508413355863403, −6.57053227690257846356724433129, −5.91776777615839210182415306177, −4.99515197435063702115655041606, −4.14825913890658922721585533390, −3.22344722537924630701057992130, −1.98368016560733583978106973890, −1.30052772714885387640001698616, 0,
1.30052772714885387640001698616, 1.98368016560733583978106973890, 3.22344722537924630701057992130, 4.14825913890658922721585533390, 4.99515197435063702115655041606, 5.91776777615839210182415306177, 6.57053227690257846356724433129, 7.38605823321025508413355863403, 8.101701226178540212510230196486
