| L(s) = 1 | + 0.144·2-s − 1.97·4-s − 4.08·5-s − 0.828·7-s − 0.574·8-s − 0.590·10-s + 0.0328·11-s − 3.12·13-s − 0.119·14-s + 3.87·16-s + 1.27·17-s − 4.11·19-s + 8.09·20-s + 0.00474·22-s + 23-s + 11.7·25-s − 0.452·26-s + 1.63·28-s + 29-s + 6.85·31-s + 1.70·32-s + 0.183·34-s + 3.38·35-s − 3.30·37-s − 0.595·38-s + 2.35·40-s + 6.64·41-s + ⋯ |
| L(s) = 1 | + 0.102·2-s − 0.989·4-s − 1.82·5-s − 0.313·7-s − 0.203·8-s − 0.186·10-s + 0.00991·11-s − 0.868·13-s − 0.0319·14-s + 0.968·16-s + 0.308·17-s − 0.945·19-s + 1.80·20-s + 0.00101·22-s + 0.208·23-s + 2.34·25-s − 0.0886·26-s + 0.309·28-s + 0.185·29-s + 1.23·31-s + 0.302·32-s + 0.0314·34-s + 0.572·35-s − 0.542·37-s − 0.0965·38-s + 0.371·40-s + 1.03·41-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 | \( 1 - 0.144T + 2T^{2} \) |
| 5 | \( 1 + 4.08T + 5T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 - 0.0328T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 - 1.27T + 17T^{2} \) |
| 19 | \( 1 + 4.11T + 19T^{2} \) |
| 31 | \( 1 - 6.85T + 31T^{2} \) |
| 37 | \( 1 + 3.30T + 37T^{2} \) |
| 41 | \( 1 - 6.64T + 41T^{2} \) |
| 43 | \( 1 + 1.68T + 43T^{2} \) |
| 47 | \( 1 - 5.69T + 47T^{2} \) |
| 53 | \( 1 - 7.40T + 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 + 4.02T + 67T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 - 2.01T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 7.36T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−7.87216568231884167634778803415, −7.16336218112022253216554700190, −6.41770708283532483731678285393, −5.33043676867115594818084153861, −4.61695281489698523808938079593, −4.12676004932706355134996166731, −3.45023289563605861629796634050, −2.61634960698360727317209060096, −0.866377066549047050069742269284, 0,
0.866377066549047050069742269284, 2.61634960698360727317209060096, 3.45023289563605861629796634050, 4.12676004932706355134996166731, 4.61695281489698523808938079593, 5.33043676867115594818084153861, 6.41770708283532483731678285393, 7.16336218112022253216554700190, 7.87216568231884167634778803415
