| L(s) = 1 | − 0.239·5-s − 7-s + 5.12·11-s − 4.88·13-s − 3.70·17-s + 3.66·19-s − 7.42·23-s − 4.94·25-s − 3.46·29-s − 0.717·31-s + 0.239·35-s + 4.60·37-s − 5.60·41-s + 12.4·43-s − 4.33·47-s + 49-s + 0.942·53-s − 1.22·55-s − 7.57·59-s + 5.50·61-s + 1.16·65-s − 0.660·67-s − 13.7·71-s − 3.66·73-s − 5.12·77-s − 6.22·79-s − 9.70·83-s + ⋯ |
| L(s) = 1 | − 0.106·5-s − 0.377·7-s + 1.54·11-s − 1.35·13-s − 0.898·17-s + 0.839·19-s − 1.54·23-s − 0.988·25-s − 0.643·29-s − 0.128·31-s + 0.0404·35-s + 0.756·37-s − 0.875·41-s + 1.90·43-s − 0.633·47-s + 0.142·49-s + 0.129·53-s − 0.165·55-s − 0.986·59-s + 0.705·61-s + 0.144·65-s − 0.0806·67-s − 1.63·71-s − 0.428·73-s − 0.584·77-s − 0.700·79-s − 1.06·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 0.239T + 5T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 - 3.66T + 19T^{2} \) |
| 23 | \( 1 + 7.42T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 0.717T + 31T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 4.33T + 47T^{2} \) |
| 53 | \( 1 - 0.942T + 53T^{2} \) |
| 59 | \( 1 + 7.57T + 59T^{2} \) |
| 61 | \( 1 - 5.50T + 61T^{2} \) |
| 67 | \( 1 + 0.660T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 3.66T + 73T^{2} \) |
| 79 | \( 1 + 6.22T + 79T^{2} \) |
| 83 | \( 1 + 9.70T + 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−8.752857055851804602153299587987, −7.73614878834370079651814229214, −7.13680425738851781114520364224, −6.30293316676175881342753178060, −5.59363285794772898477881406677, −4.39095307747203220138943377344, −3.88200947149418833363244070641, −2.67870240381194052101944133764, −1.63042204979926934887120813994, 0,
1.63042204979926934887120813994, 2.67870240381194052101944133764, 3.88200947149418833363244070641, 4.39095307747203220138943377344, 5.59363285794772898477881406677, 6.30293316676175881342753178060, 7.13680425738851781114520364224, 7.73614878834370079651814229214, 8.752857055851804602153299587987
