L(s) = 1 | + 3-s − 1.92·5-s + 2.55·7-s + 9-s + 0.815·11-s + 5.13·13-s − 1.92·15-s − 6.73·17-s − 0.147·19-s + 2.55·21-s − 9.03·23-s − 1.28·25-s + 27-s − 7.65·29-s − 3.76·31-s + 0.815·33-s − 4.91·35-s − 4.31·37-s + 5.13·39-s + 10.2·41-s + 4.82·43-s − 1.92·45-s + 2.32·47-s − 0.495·49-s − 6.73·51-s − 10.7·53-s − 1.57·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.862·5-s + 0.963·7-s + 0.333·9-s + 0.245·11-s + 1.42·13-s − 0.497·15-s − 1.63·17-s − 0.0337·19-s + 0.556·21-s − 1.88·23-s − 0.256·25-s + 0.192·27-s − 1.42·29-s − 0.676·31-s + 0.141·33-s − 0.831·35-s − 0.709·37-s + 0.822·39-s + 1.60·41-s + 0.736·43-s − 0.287·45-s + 0.339·47-s − 0.0708·49-s − 0.943·51-s − 1.47·53-s − 0.211·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 + T \) |
good | 5 | \( 1 + 1.92T + 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 - 0.815T + 11T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 17 | \( 1 + 6.73T + 17T^{2} \) |
| 19 | \( 1 + 0.147T + 19T^{2} \) |
| 23 | \( 1 + 9.03T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 3.76T + 31T^{2} \) |
| 37 | \( 1 + 4.31T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 - 2.32T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 4.54T + 59T^{2} \) |
| 61 | \( 1 - 2.14T + 61T^{2} \) |
| 67 | \( 1 + 8.28T + 67T^{2} \) |
| 71 | \( 1 + 5.89T + 71T^{2} \) |
| 73 | \( 1 + 3.28T + 73T^{2} \) |
| 79 | \( 1 - 5.54T + 79T^{2} \) |
| 83 | \( 1 + 9.06T + 83T^{2} \) |
| 89 | \( 1 + 0.924T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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
−7.84064953647951377135375625460, −7.27126382727487289746298972742, −6.27776522470788758181093124325, −5.68143580885602930402730034508, −4.39902279095080768605602442845, −4.14558259671866002209604384257, −3.42740355089944424525871035054, −2.16448588238808329773340238288, −1.54178442154159973370982534611, 0,
1.54178442154159973370982534611, 2.16448588238808329773340238288, 3.42740355089944424525871035054, 4.14558259671866002209604384257, 4.39902279095080768605602442845, 5.68143580885602930402730034508, 6.27776522470788758181093124325, 7.27126382727487289746298972742, 7.84064953647951377135375625460