L(s) = 1 | − 5-s − 1.56·7-s + 11-s + 2·13-s − 3.56·17-s + 1.56·19-s − 3.12·23-s + 25-s + 2.68·29-s − 2.43·31-s + 1.56·35-s + 6.68·37-s − 2·41-s + 6.24·43-s − 4.87·47-s − 4.56·49-s − 0.438·53-s − 55-s − 7.12·59-s + 14.6·61-s − 2·65-s + 10.2·67-s − 8.68·71-s − 2·73-s − 1.56·77-s − 9.36·79-s − 3.12·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.590·7-s + 0.301·11-s + 0.554·13-s − 0.863·17-s + 0.358·19-s − 0.651·23-s + 0.200·25-s + 0.498·29-s − 0.437·31-s + 0.263·35-s + 1.09·37-s − 0.312·41-s + 0.952·43-s − 0.711·47-s − 0.651·49-s − 0.0602·53-s − 0.134·55-s − 0.927·59-s + 1.88·61-s − 0.248·65-s + 1.25·67-s − 1.03·71-s − 0.234·73-s − 0.177·77-s − 1.05·79-s − 0.342·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 + 1.56T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 - 2.68T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 6.68T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + 4.87T + 47T^{2} \) |
| 53 | \( 1 + 0.438T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 8.68T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 9.36T + 79T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 - 8.43T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.49239849729328195465932937746, −6.72633103393073266004091860030, −6.24890522141297399451946130037, −5.46872103272161474476497025779, −4.51536497150318530782591768223, −3.92761848324211068144625572683, −3.17931771598565961749959093966, −2.31005459100045372567236358959, −1.17787551093557067709920220950, 0,
1.17787551093557067709920220950, 2.31005459100045372567236358959, 3.17931771598565961749959093966, 3.92761848324211068144625572683, 4.51536497150318530782591768223, 5.46872103272161474476497025779, 6.24890522141297399451946130037, 6.72633103393073266004091860030, 7.49239849729328195465932937746