| L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s − 8-s − 2·9-s − 3·10-s + 6·11-s − 12-s − 13-s − 3·15-s + 16-s + 3·17-s + 2·18-s − 2·19-s + 3·20-s − 6·22-s + 24-s + 4·25-s + 26-s + 5·27-s + 6·29-s + 3·30-s + 4·31-s − 32-s − 6·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.948·10-s + 1.80·11-s − 0.288·12-s − 0.277·13-s − 0.774·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.458·19-s + 0.670·20-s − 1.27·22-s + 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.962·27-s + 1.11·29-s + 0.547·30-s + 0.718·31-s − 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.315068226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315068226\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.740938448937344053696709728324, −8.946562857758601715546974072812, −8.344527680735992356525053307896, −6.96449568556298499917931526210, −6.34998887179359330478687747079, −5.79958575424076431462182408170, −4.77536500182716535947737093418, −3.32295026483676787200911474950, −2.08829293247371856153597309217, −1.02306378019630947306317213885,
1.02306378019630947306317213885, 2.08829293247371856153597309217, 3.32295026483676787200911474950, 4.77536500182716535947737093418, 5.79958575424076431462182408170, 6.34998887179359330478687747079, 6.96449568556298499917931526210, 8.344527680735992356525053307896, 8.946562857758601715546974072812, 9.740938448937344053696709728324
