| L(s) = 1 | + 0.740·2-s − 9.08·3-s − 7.45·4-s − 1.06·5-s − 6.72·6-s − 11.4·8-s + 55.4·9-s − 0.787·10-s − 37.4·11-s + 67.6·12-s + 27.1·13-s + 9.66·15-s + 51.1·16-s − 57.9·17-s + 41.0·18-s − 98.6·19-s + 7.92·20-s − 27.7·22-s − 59.0·23-s + 103.·24-s − 123.·25-s + 20.0·26-s − 258.·27-s − 230.·29-s + 7.15·30-s + 203.·31-s + 129.·32-s + ⋯ |
| L(s) = 1 | + 0.261·2-s − 1.74·3-s − 0.931·4-s − 0.0951·5-s − 0.457·6-s − 0.505·8-s + 2.05·9-s − 0.0249·10-s − 1.02·11-s + 1.62·12-s + 0.578·13-s + 0.166·15-s + 0.799·16-s − 0.827·17-s + 0.537·18-s − 1.19·19-s + 0.0886·20-s − 0.268·22-s − 0.535·23-s + 0.883·24-s − 0.990·25-s + 0.151·26-s − 1.84·27-s − 1.47·29-s + 0.0435·30-s + 1.18·31-s + 0.714·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 43 | \( 1 + 43T \) |
| good | 2 | \( 1 - 0.740T + 8T^{2} \) |
| 3 | \( 1 + 9.08T + 27T^{2} \) |
| 5 | \( 1 + 1.06T + 125T^{2} \) |
| 11 | \( 1 + 37.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 57.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 98.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 59.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 230.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 203.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 331.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 169.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 540.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 237.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 211.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 285.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 862.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 105.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 935.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 579.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 608.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 659.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 572.T + 9.12e5T^{2} \) |
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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
−8.252137045193937061556297838849, −7.56720083511725327994954315738, −6.31361956486384929648399189445, −5.98842051991126828631691216582, −5.17300000504000123487578789705, −4.43689436242186573165231599251, −3.86410243816593339942110566840, −2.21922265412846551652454011219, −0.74470332015773328993660357547, 0,
0.74470332015773328993660357547, 2.21922265412846551652454011219, 3.86410243816593339942110566840, 4.43689436242186573165231599251, 5.17300000504000123487578789705, 5.98842051991126828631691216582, 6.31361956486384929648399189445, 7.56720083511725327994954315738, 8.252137045193937061556297838849
