| L(s) = 1 | − 2·5-s + 4·7-s + 4·11-s + 13-s − 2·17-s − 25-s + 10·29-s − 4·31-s − 8·35-s − 2·37-s − 6·41-s + 12·43-s + 9·49-s − 6·53-s − 8·55-s + 12·59-s − 2·61-s − 2·65-s + 8·67-s + 2·73-s + 16·77-s − 8·79-s + 4·83-s + 4·85-s + 2·89-s + 4·91-s + 10·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 1.20·11-s + 0.277·13-s − 0.485·17-s − 1/5·25-s + 1.85·29-s − 0.718·31-s − 1.35·35-s − 0.328·37-s − 0.937·41-s + 1.82·43-s + 9/7·49-s − 0.824·53-s − 1.07·55-s + 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.977·67-s + 0.234·73-s + 1.82·77-s − 0.900·79-s + 0.439·83-s + 0.433·85-s + 0.211·89-s + 0.419·91-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.909153812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909153812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.921039436304358900231528874001, −8.484648143275568894036347393857, −7.73604447848719394517643815023, −6.99405970659997290412829823598, −6.08826329831047007406167672262, −4.94568196736683705255358366288, −4.31391709618999720844014512009, −3.55158930907206838697007657967, −2.10208151844778724001409156325, −1.00171028893857761930540835980,
1.00171028893857761930540835980, 2.10208151844778724001409156325, 3.55158930907206838697007657967, 4.31391709618999720844014512009, 4.94568196736683705255358366288, 6.08826329831047007406167672262, 6.99405970659997290412829823598, 7.73604447848719394517643815023, 8.484648143275568894036347393857, 8.921039436304358900231528874001
