| L(s) = 1 | − 2-s − 2.98·3-s + 4-s + 3.18·5-s + 2.98·6-s − 3.65·7-s − 8-s + 5.91·9-s − 3.18·10-s + 5.85·11-s − 2.98·12-s + 3.65·14-s − 9.52·15-s + 16-s − 7.18·17-s − 5.91·18-s + 19-s + 3.18·20-s + 10.9·21-s − 5.85·22-s + 6.79·23-s + 2.98·24-s + 5.17·25-s − 8.72·27-s − 3.65·28-s − 3.51·29-s + 9.52·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.72·3-s + 0.5·4-s + 1.42·5-s + 1.21·6-s − 1.38·7-s − 0.353·8-s + 1.97·9-s − 1.00·10-s + 1.76·11-s − 0.862·12-s + 0.977·14-s − 2.45·15-s + 0.250·16-s − 1.74·17-s − 1.39·18-s + 0.229·19-s + 0.713·20-s + 2.38·21-s − 1.24·22-s + 1.41·23-s + 0.609·24-s + 1.03·25-s − 1.67·27-s − 0.691·28-s − 0.652·29-s + 1.73·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 11 | \( 1 - 5.85T + 11T^{2} \) |
| 17 | \( 1 + 7.18T + 17T^{2} \) |
| 23 | \( 1 - 6.79T + 23T^{2} \) |
| 29 | \( 1 + 3.51T + 29T^{2} \) |
| 31 | \( 1 - 3.97T + 31T^{2} \) |
| 37 | \( 1 + 6.38T + 37T^{2} \) |
| 41 | \( 1 + 3.54T + 41T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 - 7.54T + 47T^{2} \) |
| 53 | \( 1 + 2.99T + 53T^{2} \) |
| 59 | \( 1 - 1.42T + 59T^{2} \) |
| 61 | \( 1 + 4.50T + 61T^{2} \) |
| 67 | \( 1 + 6.50T + 67T^{2} \) |
| 71 | \( 1 + 5.72T + 71T^{2} \) |
| 73 | \( 1 + 9.96T + 73T^{2} \) |
| 79 | \( 1 + 2.77T + 79T^{2} \) |
| 83 | \( 1 - 9.64T + 83T^{2} \) |
| 89 | \( 1 + 3.27T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−7.07774961098623762834245536736, −6.79285620969685203070274748926, −6.33397320320468799991952837763, −5.89678062655852101971914196100, −5.06009570679549675587094842155, −4.17491633759529298581975491597, −3.06957748561721768373347991048, −1.86907494058827087567662594906, −1.11956018142269595391608769339, 0,
1.11956018142269595391608769339, 1.86907494058827087567662594906, 3.06957748561721768373347991048, 4.17491633759529298581975491597, 5.06009570679549675587094842155, 5.89678062655852101971914196100, 6.33397320320468799991952837763, 6.79285620969685203070274748926, 7.07774961098623762834245536736
