L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 4·11-s − 2·13-s + 15-s − 2·17-s − 21-s + 25-s + 27-s + 6·29-s − 8·31-s − 4·33-s − 35-s + 2·37-s − 2·39-s − 6·41-s − 4·43-s + 45-s + 49-s − 2·51-s − 6·53-s − 4·55-s − 8·59-s − 2·61-s − 63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s − 0.169·35-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 0.539·55-s − 1.04·59-s − 0.256·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−8.289925822736672182905546770484, −7.52341426380507923940111350048, −6.86982879879750179525552680071, −5.98864617661575263357251261827, −5.14478606985409037175948105142, −4.43780046414078566409971573458, −3.25044325150774504833750932654, −2.62810229682407964863325346977, −1.69090786229433489075097898038, 0,
1.69090786229433489075097898038, 2.62810229682407964863325346977, 3.25044325150774504833750932654, 4.43780046414078566409971573458, 5.14478606985409037175948105142, 5.98864617661575263357251261827, 6.86982879879750179525552680071, 7.52341426380507923940111350048, 8.289925822736672182905546770484