| L(s) = 1 | − 3.94·5-s + 7-s + 0.942·11-s − 13-s + 5.60·17-s + 1.28·19-s − 4.66·23-s + 10.5·25-s − 7.66·29-s + 7.82·31-s − 3.94·35-s − 9.82·37-s + 0.942·41-s + 9.26·43-s − 5.28·47-s + 49-s − 9.22·53-s − 3.71·55-s − 9.54·59-s − 10.5·61-s + 3.94·65-s − 1.71·67-s + 3.54·71-s + 2.71·73-s + 0.942·77-s − 16.3·79-s − 0.396·83-s + ⋯ |
| L(s) = 1 | − 1.76·5-s + 0.377·7-s + 0.284·11-s − 0.277·13-s + 1.35·17-s + 0.294·19-s − 0.971·23-s + 2.10·25-s − 1.42·29-s + 1.40·31-s − 0.666·35-s − 1.61·37-s + 0.147·41-s + 1.41·43-s − 0.770·47-s + 0.142·49-s − 1.26·53-s − 0.501·55-s − 1.24·59-s − 1.35·61-s + 0.489·65-s − 0.209·67-s + 0.420·71-s + 0.318·73-s + 0.107·77-s − 1.84·79-s − 0.0435·83-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 | \( 1 + 3.94T + 5T^{2} \) |
| 11 | \( 1 - 0.942T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 + 4.66T + 23T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 + 9.82T + 37T^{2} \) |
| 41 | \( 1 - 0.942T + 41T^{2} \) |
| 43 | \( 1 - 9.26T + 43T^{2} \) |
| 47 | \( 1 + 5.28T + 47T^{2} \) |
| 53 | \( 1 + 9.22T + 53T^{2} \) |
| 59 | \( 1 + 9.54T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 1.71T + 67T^{2} \) |
| 71 | \( 1 - 3.54T + 71T^{2} \) |
| 73 | \( 1 - 2.71T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 + 0.396T + 83T^{2} \) |
| 89 | \( 1 - 5.50T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.433304626313152709862166630212, −7.69146098671606102726825698407, −7.50949303185373012243682529109, −6.37453080546484376801548883404, −5.33847236004353695316424566098, −4.46271801142382480065659893331, −3.74725483923783708536944943053, −2.98993744238960567673299943072, −1.40164097420303815630152420105, 0,
1.40164097420303815630152420105, 2.98993744238960567673299943072, 3.74725483923783708536944943053, 4.46271801142382480065659893331, 5.33847236004353695316424566098, 6.37453080546484376801548883404, 7.50949303185373012243682529109, 7.69146098671606102726825698407, 8.433304626313152709862166630212
