| L(s) = 1 | + 2-s + 4-s + 1.95·5-s + 0.971·7-s + 8-s + 1.95·10-s + 0.230·11-s − 3.30·13-s + 0.971·14-s + 16-s + 4.16·17-s + 6.27·19-s + 1.95·20-s + 0.230·22-s + 0.775·23-s − 1.18·25-s − 3.30·26-s + 0.971·28-s − 1.25·29-s + 7.09·31-s + 32-s + 4.16·34-s + 1.89·35-s − 2.16·37-s + 6.27·38-s + 1.95·40-s + 0.101·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.874·5-s + 0.367·7-s + 0.353·8-s + 0.618·10-s + 0.0695·11-s − 0.915·13-s + 0.259·14-s + 0.250·16-s + 1.00·17-s + 1.43·19-s + 0.437·20-s + 0.0492·22-s + 0.161·23-s − 0.236·25-s − 0.647·26-s + 0.183·28-s − 0.233·29-s + 1.27·31-s + 0.176·32-s + 0.713·34-s + 0.320·35-s − 0.355·37-s + 1.01·38-s + 0.309·40-s + 0.0158·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.070069885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.070069885\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 5 | \( 1 - 1.95T + 5T^{2} \) |
| 7 | \( 1 - 0.971T + 7T^{2} \) |
| 11 | \( 1 - 0.230T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 19 | \( 1 - 6.27T + 19T^{2} \) |
| 23 | \( 1 - 0.775T + 23T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 - 7.09T + 31T^{2} \) |
| 37 | \( 1 + 2.16T + 37T^{2} \) |
| 41 | \( 1 - 0.101T + 41T^{2} \) |
| 43 | \( 1 + 4.49T + 43T^{2} \) |
| 47 | \( 1 - 6.33T + 47T^{2} \) |
| 53 | \( 1 + 5.39T + 53T^{2} \) |
| 59 | \( 1 + 5.68T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 0.362T + 71T^{2} \) |
| 73 | \( 1 - 6.41T + 73T^{2} \) |
| 79 | \( 1 - 2.16T + 79T^{2} \) |
| 83 | \( 1 - 5.16T + 83T^{2} \) |
| 89 | \( 1 + 9.75T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.152398585471613886375690379879, −7.59477204337018604654977051917, −6.81099098289631174122067604571, −6.03711691719292261276393863245, −5.23921909715736035970535320233, −4.94989792435988987899925316795, −3.76786814137381114876025233296, −2.93897774400695017140001487254, −2.09058436473679784995556583134, −1.09471953545226389335134056670,
1.09471953545226389335134056670, 2.09058436473679784995556583134, 2.93897774400695017140001487254, 3.76786814137381114876025233296, 4.94989792435988987899925316795, 5.23921909715736035970535320233, 6.03711691719292261276393863245, 6.81099098289631174122067604571, 7.59477204337018604654977051917, 8.152398585471613886375690379879
