| L(s) = 1 | + 1.75·2-s + 0.604·3-s + 1.06·4-s − 5-s + 1.05·6-s − 1.98·7-s − 1.63·8-s − 2.63·9-s − 1.75·10-s + 4.62·11-s + 0.645·12-s + 1.49·13-s − 3.47·14-s − 0.604·15-s − 4.99·16-s − 0.0579·17-s − 4.61·18-s − 6.32·19-s − 1.06·20-s − 1.19·21-s + 8.10·22-s − 2.71·23-s − 0.988·24-s + 25-s + 2.62·26-s − 3.40·27-s − 2.11·28-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.349·3-s + 0.533·4-s − 0.447·5-s + 0.432·6-s − 0.749·7-s − 0.577·8-s − 0.878·9-s − 0.553·10-s + 1.39·11-s + 0.186·12-s + 0.415·13-s − 0.928·14-s − 0.156·15-s − 1.24·16-s − 0.0140·17-s − 1.08·18-s − 1.45·19-s − 0.238·20-s − 0.261·21-s + 1.72·22-s − 0.566·23-s − 0.201·24-s + 0.200·25-s + 0.514·26-s − 0.655·27-s − 0.399·28-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 | \( 1 - 1.75T + 2T^{2} \) |
| 3 | \( 1 - 0.604T + 3T^{2} \) |
| 7 | \( 1 + 1.98T + 7T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 - 1.49T + 13T^{2} \) |
| 17 | \( 1 + 0.0579T + 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 + 2.71T + 23T^{2} \) |
| 29 | \( 1 + 3.22T + 29T^{2} \) |
| 31 | \( 1 + 1.11T + 31T^{2} \) |
| 37 | \( 1 + 8.23T + 37T^{2} \) |
| 41 | \( 1 - 2.09T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 4.83T + 47T^{2} \) |
| 53 | \( 1 - 7.80T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 2.32T + 61T^{2} \) |
| 67 | \( 1 - 3.57T + 67T^{2} \) |
| 71 | \( 1 - 3.42T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 4.50T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−8.927654784831311697670389786307, −8.054186233279898328716347360963, −6.75987179799912743014431704979, −6.33987962028641578937739634029, −5.56737273002418496169077795769, −4.46410481900380362875662110878, −3.70233538624644657091896197242, −3.27406978255834388405745532434, −2.04074997549911250139950663682, 0,
2.04074997549911250139950663682, 3.27406978255834388405745532434, 3.70233538624644657091896197242, 4.46410481900380362875662110878, 5.56737273002418496169077795769, 6.33987962028641578937739634029, 6.75987179799912743014431704979, 8.054186233279898328716347360963, 8.927654784831311697670389786307
