L(s) = 1 | − 2.72·2-s + 5.42·4-s + 5-s − 4.89·7-s − 9.34·8-s − 2.72·10-s − 3.91·11-s + 5.83·13-s + 13.3·14-s + 14.6·16-s − 3.31·17-s + 1.15·19-s + 5.42·20-s + 10.6·22-s − 6.76·23-s + 25-s − 15.9·26-s − 26.5·28-s + 7.32·29-s − 0.0507·31-s − 21.1·32-s + 9.04·34-s − 4.89·35-s + 2.18·37-s − 3.13·38-s − 9.34·40-s + 10.1·41-s + ⋯ |
L(s) = 1 | − 1.92·2-s + 2.71·4-s + 0.447·5-s − 1.85·7-s − 3.30·8-s − 0.861·10-s − 1.18·11-s + 1.61·13-s + 3.56·14-s + 3.65·16-s − 0.805·17-s + 0.264·19-s + 1.21·20-s + 2.27·22-s − 1.41·23-s + 0.200·25-s − 3.11·26-s − 5.02·28-s + 1.36·29-s − 0.00911·31-s − 3.73·32-s + 1.55·34-s − 0.827·35-s + 0.358·37-s − 0.508·38-s − 1.47·40-s + 1.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 + 3.91T + 11T^{2} \) |
| 13 | \( 1 - 5.83T + 13T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 19 | \( 1 - 1.15T + 19T^{2} \) |
| 23 | \( 1 + 6.76T + 23T^{2} \) |
| 29 | \( 1 - 7.32T + 29T^{2} \) |
| 31 | \( 1 + 0.0507T + 31T^{2} \) |
| 37 | \( 1 - 2.18T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 2.07T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 5.88T + 53T^{2} \) |
| 59 | \( 1 + 2.22T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 2.18T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 3.24T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 97 | \( 1 - 5.56T + 97T^{2} \) |
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
−8.178635129702177877158126636560, −7.61263418749745680588125807306, −6.57909578935374782117157758821, −6.28222968595891350814483710887, −5.69575170994713190353309580500, −3.90856225003866304186272086100, −2.87593894209009403093814497706, −2.38025797074112799315393003757, −1.03922923542518550141444418598, 0,
1.03922923542518550141444418598, 2.38025797074112799315393003757, 2.87593894209009403093814497706, 3.90856225003866304186272086100, 5.69575170994713190353309580500, 6.28222968595891350814483710887, 6.57909578935374782117157758821, 7.61263418749745680588125807306, 8.178635129702177877158126636560