| L(s) = 1 | + 3-s − 5-s − 3.74·7-s + 9-s − 5.48·11-s − 1.32·13-s − 15-s + 6.95·17-s + 5.06·19-s − 3.74·21-s + 2.11·23-s + 25-s + 27-s + 3.63·29-s + 31-s − 5.48·33-s + 3.74·35-s + 8.39·37-s − 1.32·39-s + 11.4·41-s − 0.421·43-s − 45-s + 1.46·47-s + 7.02·49-s + 6.95·51-s + 4.53·53-s + 5.48·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.41·7-s + 0.333·9-s − 1.65·11-s − 0.367·13-s − 0.258·15-s + 1.68·17-s + 1.16·19-s − 0.817·21-s + 0.440·23-s + 0.200·25-s + 0.192·27-s + 0.674·29-s + 0.179·31-s − 0.955·33-s + 0.632·35-s + 1.37·37-s − 0.211·39-s + 1.79·41-s − 0.0642·43-s − 0.149·45-s + 0.213·47-s + 1.00·49-s + 0.973·51-s + 0.622·53-s + 0.740·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.496087548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496087548\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 + 5.48T + 11T^{2} \) |
| 13 | \( 1 + 1.32T + 13T^{2} \) |
| 17 | \( 1 - 6.95T + 17T^{2} \) |
| 19 | \( 1 - 5.06T + 19T^{2} \) |
| 23 | \( 1 - 2.11T + 23T^{2} \) |
| 29 | \( 1 - 3.63T + 29T^{2} \) |
| 37 | \( 1 - 8.39T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 0.421T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 - 4.78T + 59T^{2} \) |
| 61 | \( 1 - 6.84T + 61T^{2} \) |
| 67 | \( 1 + 4.39T + 67T^{2} \) |
| 71 | \( 1 + 7.34T + 71T^{2} \) |
| 73 | \( 1 + 4.39T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 5.04T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 3.48T + 97T^{2} \) |
| 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
−9.403662159863700202996624845818, −8.339243952400910914170374315865, −7.59574572637672608177893981276, −7.20012535999822078892878140147, −5.95391610656811717315150677358, −5.27373495959317716525241124937, −4.11661984247401183516745390275, −2.94070378935393994258501923150, −2.84728348437911541135430721737, −0.791459501268609649728251586996,
0.791459501268609649728251586996, 2.84728348437911541135430721737, 2.94070378935393994258501923150, 4.11661984247401183516745390275, 5.27373495959317716525241124937, 5.95391610656811717315150677358, 7.20012535999822078892878140147, 7.59574572637672608177893981276, 8.339243952400910914170374315865, 9.403662159863700202996624845818
