| L(s) = 1 | − 0.706·2-s + 2.56·3-s − 1.50·4-s − 5-s − 1.81·6-s − 2.27·7-s + 2.47·8-s + 3.56·9-s + 0.706·10-s − 1.04·11-s − 3.84·12-s − 1.52·13-s + 1.60·14-s − 2.56·15-s + 1.25·16-s − 1.78·17-s − 2.51·18-s + 5.84·19-s + 1.50·20-s − 5.82·21-s + 0.737·22-s + 4.21·23-s + 6.33·24-s + 25-s + 1.07·26-s + 1.44·27-s + 3.41·28-s + ⋯ |
| L(s) = 1 | − 0.499·2-s + 1.47·3-s − 0.750·4-s − 0.447·5-s − 0.739·6-s − 0.859·7-s + 0.874·8-s + 1.18·9-s + 0.223·10-s − 0.314·11-s − 1.10·12-s − 0.423·13-s + 0.429·14-s − 0.661·15-s + 0.312·16-s − 0.432·17-s − 0.593·18-s + 1.34·19-s + 0.335·20-s − 1.27·21-s + 0.157·22-s + 0.879·23-s + 1.29·24-s + 0.200·25-s + 0.211·26-s + 0.278·27-s + 0.644·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.706T + 2T^{2} \) |
| 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 + 2.27T + 7T^{2} \) |
| 11 | \( 1 + 1.04T + 11T^{2} \) |
| 13 | \( 1 + 1.52T + 13T^{2} \) |
| 17 | \( 1 + 1.78T + 17T^{2} \) |
| 19 | \( 1 - 5.84T + 19T^{2} \) |
| 23 | \( 1 - 4.21T + 23T^{2} \) |
| 29 | \( 1 + 3.31T + 29T^{2} \) |
| 31 | \( 1 + 9.00T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 5.86T + 43T^{2} \) |
| 47 | \( 1 + 6.85T + 47T^{2} \) |
| 53 | \( 1 + 6.17T + 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 + 5.88T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 1.46T + 79T^{2} \) |
| 83 | \( 1 - 5.14T + 83T^{2} \) |
| 89 | \( 1 - 6.89T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.919515389229184704839379486057, −8.003281743213569929129668200563, −7.60469853762694879229217455239, −6.79456238952737256047912338455, −5.38470262138972861052727501815, −4.49879990470283990634882827652, −3.45077833113885382864202058808, −3.03961456579007080952882924606, −1.63781189060551379846166277134, 0,
1.63781189060551379846166277134, 3.03961456579007080952882924606, 3.45077833113885382864202058808, 4.49879990470283990634882827652, 5.38470262138972861052727501815, 6.79456238952737256047912338455, 7.60469853762694879229217455239, 8.003281743213569929129668200563, 8.919515389229184704839379486057
