| L(s) = 1 | + 1.97·2-s + 1.88·4-s − 1.48·5-s + 2.73·7-s − 0.233·8-s − 2.92·10-s − 11-s + 2.15·13-s + 5.38·14-s − 4.22·16-s + 1.06·17-s − 6.24·19-s − 2.79·20-s − 1.97·22-s − 5.92·23-s − 2.79·25-s + 4.23·26-s + 5.14·28-s + 3.70·29-s − 1.98·31-s − 7.85·32-s + 2.09·34-s − 4.05·35-s − 11.5·37-s − 12.3·38-s + 0.346·40-s + 4.94·41-s + ⋯ |
| L(s) = 1 | + 1.39·2-s + 0.940·4-s − 0.663·5-s + 1.03·7-s − 0.0824·8-s − 0.924·10-s − 0.301·11-s + 0.596·13-s + 1.43·14-s − 1.05·16-s + 0.257·17-s − 1.43·19-s − 0.624·20-s − 0.420·22-s − 1.23·23-s − 0.559·25-s + 0.831·26-s + 0.971·28-s + 0.687·29-s − 0.356·31-s − 1.38·32-s + 0.358·34-s − 0.685·35-s − 1.89·37-s − 1.99·38-s + 0.0547·40-s + 0.771·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.97T + 2T^{2} \) |
| 5 | \( 1 + 1.48T + 5T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 13 | \( 1 - 2.15T + 13T^{2} \) |
| 17 | \( 1 - 1.06T + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 23 | \( 1 + 5.92T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + 1.98T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 4.94T + 41T^{2} \) |
| 43 | \( 1 - 8.44T + 43T^{2} \) |
| 47 | \( 1 + 8.23T + 47T^{2} \) |
| 53 | \( 1 - 3.86T + 53T^{2} \) |
| 59 | \( 1 + 5.17T + 59T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 3.27T + 71T^{2} \) |
| 73 | \( 1 + 0.437T + 73T^{2} \) |
| 79 | \( 1 - 0.577T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 5.54T + 89T^{2} \) |
| 97 | \( 1 - 0.0714T + 97T^{2} \) |
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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
−7.70954771549420577844162729876, −6.81933569514425778569303643540, −6.07516826879436131776947534730, −5.46373587635500481063705254657, −4.67410392592519390513797578898, −4.11083092595377357197268995939, −3.59138430258728914106065233981, −2.51406857904359900978698042018, −1.67803998072461462223970974678, 0,
1.67803998072461462223970974678, 2.51406857904359900978698042018, 3.59138430258728914106065233981, 4.11083092595377357197268995939, 4.67410392592519390513797578898, 5.46373587635500481063705254657, 6.07516826879436131776947534730, 6.81933569514425778569303643540, 7.70954771549420577844162729876
