L(s) = 1 | + 0.941·2-s − 2.58·3-s − 1.11·4-s − 5-s − 2.43·6-s + 7-s − 2.93·8-s + 3.70·9-s − 0.941·10-s − 4.45·11-s + 2.88·12-s + 0.279·13-s + 0.941·14-s + 2.58·15-s − 0.531·16-s − 4.16·17-s + 3.48·18-s − 1.13·19-s + 1.11·20-s − 2.58·21-s − 4.19·22-s + 0.849·23-s + 7.58·24-s + 25-s + 0.262·26-s − 1.81·27-s − 1.11·28-s + ⋯ |
L(s) = 1 | + 0.665·2-s − 1.49·3-s − 0.556·4-s − 0.447·5-s − 0.994·6-s + 0.377·7-s − 1.03·8-s + 1.23·9-s − 0.297·10-s − 1.34·11-s + 0.832·12-s + 0.0774·13-s + 0.251·14-s + 0.668·15-s − 0.132·16-s − 1.01·17-s + 0.821·18-s − 0.259·19-s + 0.249·20-s − 0.564·21-s − 0.893·22-s + 0.177·23-s + 1.54·24-s + 0.200·25-s + 0.0515·26-s − 0.349·27-s − 0.210·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 - 0.941T + 2T^{2} \) |
| 3 | \( 1 + 2.58T + 3T^{2} \) |
| 11 | \( 1 + 4.45T + 11T^{2} \) |
| 13 | \( 1 - 0.279T + 13T^{2} \) |
| 17 | \( 1 + 4.16T + 17T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 23 | \( 1 - 0.849T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 - 3.93T + 31T^{2} \) |
| 37 | \( 1 - 7.47T + 37T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 + 7.00T + 53T^{2} \) |
| 59 | \( 1 - 2.38T + 59T^{2} \) |
| 61 | \( 1 - 3.10T + 61T^{2} \) |
| 67 | \( 1 - 3.10T + 67T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 + 1.67T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 0.697T + 89T^{2} \) |
| 97 | \( 1 + 1.55T + 97T^{2} \) |
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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
−7.29820394164098313299162633837, −6.51619501364993480937223104964, −5.97310780787905323948803853778, −5.21863509615927315163293993631, −4.74171347397676039170147211225, −4.36595993487917664483062448307, −3.28397344721552954831843700713, −2.35810364711106540274871946217, −0.839979138427191763634260674268, 0,
0.839979138427191763634260674268, 2.35810364711106540274871946217, 3.28397344721552954831843700713, 4.36595993487917664483062448307, 4.74171347397676039170147211225, 5.21863509615927315163293993631, 5.97310780787905323948803853778, 6.51619501364993480937223104964, 7.29820394164098313299162633837