L(s) = 1 | − 0.563·2-s − 3-s − 1.68·4-s + 0.563·6-s + 1.09·7-s + 2.07·8-s + 9-s + 0.827·11-s + 1.68·12-s + 2.66·13-s − 0.617·14-s + 2.19·16-s − 4.16·17-s − 0.563·18-s − 5.48·19-s − 1.09·21-s − 0.466·22-s − 5.61·23-s − 2.07·24-s − 1.50·26-s − 27-s − 1.84·28-s + 9.79·29-s − 9.33·31-s − 5.38·32-s − 0.827·33-s + 2.34·34-s + ⋯ |
L(s) = 1 | − 0.398·2-s − 0.577·3-s − 0.841·4-s + 0.230·6-s + 0.414·7-s + 0.733·8-s + 0.333·9-s + 0.249·11-s + 0.485·12-s + 0.738·13-s − 0.164·14-s + 0.549·16-s − 1.01·17-s − 0.132·18-s − 1.25·19-s − 0.239·21-s − 0.0993·22-s − 1.17·23-s − 0.423·24-s − 0.294·26-s − 0.192·27-s − 0.348·28-s + 1.81·29-s − 1.67·31-s − 0.952·32-s − 0.144·33-s + 0.402·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + 0.563T + 2T^{2} \) |
| 7 | \( 1 - 1.09T + 7T^{2} \) |
| 11 | \( 1 - 0.827T + 11T^{2} \) |
| 13 | \( 1 - 2.66T + 13T^{2} \) |
| 17 | \( 1 + 4.16T + 17T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 + 5.61T + 23T^{2} \) |
| 29 | \( 1 - 9.79T + 29T^{2} \) |
| 31 | \( 1 + 9.33T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 - 2.92T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 + 9.17T + 59T^{2} \) |
| 61 | \( 1 + 4.17T + 61T^{2} \) |
| 67 | \( 1 - 7.79T + 67T^{2} \) |
| 71 | \( 1 + 8.28T + 71T^{2} \) |
| 73 | \( 1 - 4.51T + 73T^{2} \) |
| 79 | \( 1 - 3.96T + 79T^{2} \) |
| 83 | \( 1 - 9.53T + 83T^{2} \) |
| 89 | \( 1 - 4.86T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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
−8.294643934225786296775225172325, −7.67000393869569052324212284505, −6.56211315836498809328328035311, −6.06702187646735370933410773819, −5.06233789952297257544039636106, −4.33714052239262040782475743505, −3.84012221021171438264690283833, −2.27242840159609778459365468046, −1.20445986569006262176566468361, 0,
1.20445986569006262176566468361, 2.27242840159609778459365468046, 3.84012221021171438264690283833, 4.33714052239262040782475743505, 5.06233789952297257544039636106, 6.06702187646735370933410773819, 6.56211315836498809328328035311, 7.67000393869569052324212284505, 8.294643934225786296775225172325