| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s + 4·11-s + 13-s + 2·15-s − 2·17-s + 4·19-s − 21-s + 4·23-s − 25-s − 27-s − 2·29-s − 4·33-s − 2·35-s − 2·37-s − 39-s + 2·41-s − 4·43-s − 2·45-s + 12·47-s + 49-s + 2·51-s + 6·53-s − 8·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.516·15-s − 0.485·17-s + 0.917·19-s − 0.218·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s − 0.338·35-s − 0.328·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s + 1.75·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 1.07·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.435702702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435702702\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−8.412750938401679695200426347506, −7.39338686261700899727743769729, −7.11048801257058618698567996372, −6.15136398849261121003233660792, −5.44236676719334468599667964443, −4.49384158917216273081012753018, −3.97402825163766103093059260610, −3.10287718805092059131516160479, −1.70740221809429506918907016767, −0.72803607004409955458364897150,
0.72803607004409955458364897150, 1.70740221809429506918907016767, 3.10287718805092059131516160479, 3.97402825163766103093059260610, 4.49384158917216273081012753018, 5.44236676719334468599667964443, 6.15136398849261121003233660792, 7.11048801257058618698567996372, 7.39338686261700899727743769729, 8.412750938401679695200426347506
