| L(s) = 1 | − 1.39·2-s − 0.0545·4-s + 2.50·5-s + 4.61·7-s + 2.86·8-s − 3.49·10-s − 6.41·11-s − 4.63·13-s − 6.44·14-s − 3.88·16-s + 0.986·17-s − 2.31·19-s − 0.136·20-s + 8.94·22-s + 23-s + 1.26·25-s + 6.45·26-s − 0.252·28-s − 29-s + 7.37·31-s − 0.308·32-s − 1.37·34-s + 11.5·35-s + 4.51·37-s + 3.23·38-s + 7.17·40-s − 9.39·41-s + ⋯ |
| L(s) = 1 | − 0.986·2-s − 0.0272·4-s + 1.11·5-s + 1.74·7-s + 1.01·8-s − 1.10·10-s − 1.93·11-s − 1.28·13-s − 1.72·14-s − 0.971·16-s + 0.239·17-s − 0.532·19-s − 0.0305·20-s + 1.90·22-s + 0.208·23-s + 0.252·25-s + 1.26·26-s − 0.0476·28-s − 0.185·29-s + 1.32·31-s − 0.0545·32-s − 0.235·34-s + 1.95·35-s + 0.742·37-s + 0.524·38-s + 1.13·40-s − 1.46·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.39T + 2T^{2} \) |
| 5 | \( 1 - 2.50T + 5T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 + 6.41T + 11T^{2} \) |
| 13 | \( 1 + 4.63T + 13T^{2} \) |
| 17 | \( 1 - 0.986T + 17T^{2} \) |
| 19 | \( 1 + 2.31T + 19T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 - 4.51T + 37T^{2} \) |
| 41 | \( 1 + 9.39T + 41T^{2} \) |
| 43 | \( 1 - 0.107T + 43T^{2} \) |
| 47 | \( 1 + 0.665T + 47T^{2} \) |
| 53 | \( 1 + 9.08T + 53T^{2} \) |
| 59 | \( 1 + 5.42T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 1.36T + 73T^{2} \) |
| 79 | \( 1 - 4.07T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 3.27T + 89T^{2} \) |
| 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
−7.72318537953235423420120743314, −7.57959067318193625073499521214, −6.36984233529104954212977604062, −5.24507259248966923582327075627, −5.06316748444123139634042842117, −4.41937793887378542134485616604, −2.70733958323281084289593463827, −2.11816762624023058335145062658, −1.35367276417925811473023442763, 0,
1.35367276417925811473023442763, 2.11816762624023058335145062658, 2.70733958323281084289593463827, 4.41937793887378542134485616604, 5.06316748444123139634042842117, 5.24507259248966923582327075627, 6.36984233529104954212977604062, 7.57959067318193625073499521214, 7.72318537953235423420120743314
