| L(s) = 1 | − 3.37·3-s + 5-s + 8.37·9-s − 0.627·11-s + 1.37·13-s − 3.37·15-s − 5.37·17-s + 6.74·19-s − 6.74·23-s + 25-s − 18.1·27-s + 1.37·29-s − 8·31-s + 2.11·33-s − 2·37-s − 4.62·39-s + 4.74·41-s − 2.74·43-s + 8.37·45-s + 10.1·47-s + 18.1·51-s − 0.744·53-s − 0.627·55-s − 22.7·57-s + 8·59-s − 8.74·61-s + 1.37·65-s + ⋯ |
| L(s) = 1 | − 1.94·3-s + 0.447·5-s + 2.79·9-s − 0.189·11-s + 0.380·13-s − 0.870·15-s − 1.30·17-s + 1.54·19-s − 1.40·23-s + 0.200·25-s − 3.48·27-s + 0.254·29-s − 1.43·31-s + 0.368·33-s − 0.328·37-s − 0.741·39-s + 0.740·41-s − 0.418·43-s + 1.24·45-s + 1.47·47-s + 2.53·51-s − 0.102·53-s − 0.0846·55-s − 3.01·57-s + 1.04·59-s − 1.11·61-s + 0.170·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.37T + 3T^{2} \) |
| 11 | \( 1 + 0.627T + 11T^{2} \) |
| 13 | \( 1 - 1.37T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 + 6.74T + 23T^{2} \) |
| 29 | \( 1 - 1.37T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.74T + 41T^{2} \) |
| 43 | \( 1 + 2.74T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 0.744T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 8.74T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 2.11T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
| 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
−7.84203182632699118213000383556, −7.11751261256207745575014903405, −6.49490272768468117134143742294, −5.74576289682242546573333651805, −5.36873739608471021665711810954, −4.49511260737420677607847196876, −3.74053866015848729635518726844, −2.17012654594016786844304897496, −1.17694073576242688207767912336, 0,
1.17694073576242688207767912336, 2.17012654594016786844304897496, 3.74053866015848729635518726844, 4.49511260737420677607847196876, 5.36873739608471021665711810954, 5.74576289682242546573333651805, 6.49490272768468117134143742294, 7.11751261256207745575014903405, 7.84203182632699118213000383556
