L(s) = 1 | − 2-s + 4-s + 3.46·5-s − 2.46·7-s − 8-s − 3.46·10-s − 1.26·11-s + 5.46·13-s + 2.46·14-s + 16-s + 6.73·17-s − 5.46·19-s + 3.46·20-s + 1.26·22-s + 4.26·23-s + 6.99·25-s − 5.46·26-s − 2.46·28-s − 1.73·29-s + 31-s − 32-s − 6.73·34-s − 8.53·35-s + 2.73·37-s + 5.46·38-s − 3.46·40-s + 1.19·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.54·5-s − 0.931·7-s − 0.353·8-s − 1.09·10-s − 0.382·11-s + 1.51·13-s + 0.658·14-s + 0.250·16-s + 1.63·17-s − 1.25·19-s + 0.774·20-s + 0.270·22-s + 0.889·23-s + 1.39·25-s − 1.07·26-s − 0.465·28-s − 0.321·29-s + 0.179·31-s − 0.176·32-s − 1.15·34-s − 1.44·35-s + 0.449·37-s + 0.886·38-s − 0.547·40-s + 0.186·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.069912304\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.069912304\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 - 4.26T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 - 1.19T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 - 7.46T + 47T^{2} \) |
| 53 | \( 1 - 6.26T + 53T^{2} \) |
| 59 | \( 1 + 3.26T + 59T^{2} \) |
| 61 | \( 1 - 2.73T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 1.46T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 1.80T + 83T^{2} \) |
| 89 | \( 1 + 4.26T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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
−7.86994813606706874583501118610, −7.11399693968654131738831987883, −6.29345017517326921079363974109, −5.93283341794444826713330790608, −5.46362903217079051726161392262, −4.19131958713097240294888518133, −3.20735363672090928232361934746, −2.60165568500008787036826595074, −1.62010086539031180816470320116, −0.835592154799277896367630077198,
0.835592154799277896367630077198, 1.62010086539031180816470320116, 2.60165568500008787036826595074, 3.20735363672090928232361934746, 4.19131958713097240294888518133, 5.46362903217079051726161392262, 5.93283341794444826713330790608, 6.29345017517326921079363974109, 7.11399693968654131738831987883, 7.86994813606706874583501118610