| L(s) = 1 | + 3-s − 4·5-s − 2·7-s + 9-s − 4·11-s + 2·13-s − 4·15-s − 2·17-s − 8·19-s − 2·21-s − 4·23-s + 11·25-s + 27-s + 6·31-s − 4·33-s + 8·35-s − 2·37-s + 2·39-s + 6·41-s − 4·45-s − 4·47-s − 3·49-s − 2·51-s + 16·55-s − 8·57-s + 4·59-s + 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 1.03·15-s − 0.485·17-s − 1.83·19-s − 0.436·21-s − 0.834·23-s + 11/5·25-s + 0.192·27-s + 1.07·31-s − 0.696·33-s + 1.35·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.596·45-s − 0.583·47-s − 3/7·49-s − 0.280·51-s + 2.15·55-s − 1.05·57-s + 0.520·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.86660959821522152324936701040, −10.10337746972486500130129788437, −8.649034097305583887898795668373, −8.230854059399453444196153124887, −7.30623361332829545195635303299, −6.27798985919626463740519529757, −4.56007962912554840821971374527, −3.77214318657172177768570145308, −2.65190583901467307643981603895, 0,
2.65190583901467307643981603895, 3.77214318657172177768570145308, 4.56007962912554840821971374527, 6.27798985919626463740519529757, 7.30623361332829545195635303299, 8.230854059399453444196153124887, 8.649034097305583887898795668373, 10.10337746972486500130129788437, 10.86660959821522152324936701040
