| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 11-s − 6·13-s + 15-s + 2·17-s + 21-s + 6·23-s + 25-s − 27-s + 6·29-s − 2·31-s − 33-s + 35-s + 10·37-s + 6·39-s − 8·41-s − 8·43-s − 45-s + 4·47-s + 49-s − 2·51-s + 6·53-s − 55-s − 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s + 0.258·15-s + 0.485·17-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 + 1.64·37-s + 0.960·39-s − 1.24·41-s − 1.21·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.134·55-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−7.81993280421946382945106801724, −7.11121092576963193957274002281, −6.67325430102917671707896598298, −5.69729426126696011738753652683, −4.93698451750088092778592074564, −4.38618859180939581022517956349, −3.29788050154397325858334845898, −2.53986098106438213600381632263, −1.16962844533833673422814210249, 0,
1.16962844533833673422814210249, 2.53986098106438213600381632263, 3.29788050154397325858334845898, 4.38618859180939581022517956349, 4.93698451750088092778592074564, 5.69729426126696011738753652683, 6.67325430102917671707896598298, 7.11121092576963193957274002281, 7.81993280421946382945106801724
