| L(s) = 1 | − 0.554·3-s + 3.04·7-s − 2.69·9-s + 2.93·11-s + 3.24·13-s − 2.15·17-s + 19-s − 1.69·21-s − 1.19·23-s + 3.15·27-s − 1.77·29-s + 9.34·31-s − 1.63·33-s − 1.15·37-s − 1.80·39-s + 8.57·41-s − 5.27·43-s − 2.35·47-s + 2.29·49-s + 1.19·51-s + 8.82·53-s − 0.554·57-s + 5.70·59-s − 9.96·61-s − 8.20·63-s − 4.98·67-s + 0.664·69-s + ⋯ |
| L(s) = 1 | − 0.320·3-s + 1.15·7-s − 0.897·9-s + 0.886·11-s + 0.900·13-s − 0.523·17-s + 0.229·19-s − 0.369·21-s − 0.249·23-s + 0.607·27-s − 0.329·29-s + 1.67·31-s − 0.283·33-s − 0.190·37-s − 0.288·39-s + 1.33·41-s − 0.804·43-s − 0.343·47-s + 0.327·49-s + 0.167·51-s + 1.21·53-s − 0.0735·57-s + 0.742·59-s − 1.27·61-s − 1.03·63-s − 0.608·67-s + 0.0800·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.189996061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.189996061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.554T + 3T^{2} \) |
| 7 | \( 1 - 3.04T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 - 3.24T + 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 23 | \( 1 + 1.19T + 23T^{2} \) |
| 29 | \( 1 + 1.77T + 29T^{2} \) |
| 31 | \( 1 - 9.34T + 31T^{2} \) |
| 37 | \( 1 + 1.15T + 37T^{2} \) |
| 41 | \( 1 - 8.57T + 41T^{2} \) |
| 43 | \( 1 + 5.27T + 43T^{2} \) |
| 47 | \( 1 + 2.35T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 - 5.70T + 59T^{2} \) |
| 61 | \( 1 + 9.96T + 61T^{2} \) |
| 67 | \( 1 + 4.98T + 67T^{2} \) |
| 71 | \( 1 + 2.70T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 5.66T + 79T^{2} \) |
| 83 | \( 1 - 3.00T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 3.24T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.046670705588876678895650806700, −7.15961037254037534415493376365, −6.32077107874576219785886952820, −5.90380050183684784743871002818, −5.03039738505178739233549623149, −4.40876385437600541614968126639, −3.62311385620139522914617396624, −2.64509794167290293650236161420, −1.67138836159170516726964049936, −0.793792072834445064659240540965,
0.793792072834445064659240540965, 1.67138836159170516726964049936, 2.64509794167290293650236161420, 3.62311385620139522914617396624, 4.40876385437600541614968126639, 5.03039738505178739233549623149, 5.90380050183684784743871002818, 6.32077107874576219785886952820, 7.15961037254037534415493376365, 8.046670705588876678895650806700
