| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 11-s − 2·13-s + 15-s − 6·17-s + 8·19-s − 21-s + 6·23-s + 25-s + 27-s − 6·29-s − 2·31-s − 33-s − 35-s − 2·37-s − 2·39-s + 8·43-s + 45-s + 12·47-s + 49-s − 6·51-s − 6·53-s − 55-s + 8·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.258·15-s − 1.45·17-s + 1.83·19-s − 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.174·33-s − 0.169·35-s − 0.328·37-s − 0.320·39-s + 1.21·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 0.134·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.817187137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.817187137\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.91032280180593, −13.58367810282828, −13.23518448407240, −12.66959977159145, −12.21948869863591, −11.56483075534104, −10.92701875965485, −10.66346018897637, −9.892627551662255, −9.375380719045634, −9.149381024515461, −8.687697588108926, −7.824796940343967, −7.282672979934649, −7.113456727989229, −6.341591587220580, −5.627732096274424, −5.235804778617028, −4.562298948527812, −3.944791262483112, −3.195641611339992, −2.732383815942905, −2.164751047252680, −1.390901181366284, −0.5323062087506254,
0.5323062087506254, 1.390901181366284, 2.164751047252680, 2.732383815942905, 3.195641611339992, 3.944791262483112, 4.562298948527812, 5.235804778617028, 5.627732096274424, 6.341591587220580, 7.113456727989229, 7.282672979934649, 7.824796940343967, 8.687697588108926, 9.149381024515461, 9.375380719045634, 9.892627551662255, 10.66346018897637, 10.92701875965485, 11.56483075534104, 12.21948869863591, 12.66959977159145, 13.23518448407240, 13.58367810282828, 13.91032280180593
