| L(s) = 1 | + 3-s + 2·7-s + 9-s + 3·11-s − 5·13-s − 5·17-s + 2·21-s + 27-s − 2·29-s + 8·31-s + 3·33-s + 4·37-s − 5·39-s + 10·41-s + 43-s − 3·47-s − 3·49-s − 5·51-s − 2·53-s − 9·59-s − 4·61-s + 2·63-s + 13·67-s + 6·71-s − 8·73-s + 6·77-s + 5·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.904·11-s − 1.38·13-s − 1.21·17-s + 0.436·21-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.522·33-s + 0.657·37-s − 0.800·39-s + 1.56·41-s + 0.152·43-s − 0.437·47-s − 3/7·49-s − 0.700·51-s − 0.274·53-s − 1.17·59-s − 0.512·61-s + 0.251·63-s + 1.58·67-s + 0.712·71-s − 0.936·73-s + 0.683·77-s + 0.562·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.137883116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.137883116\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.35073320452132, −14.18046090678456, −13.59102018218276, −12.91076493928445, −12.50362544334486, −11.91737823663184, −11.37615591573045, −11.01719976917835, −10.29001460372594, −9.668775549366504, −9.290708119978733, −8.835556730971857, −8.051245871883114, −7.814919782442171, −7.118944047595196, −6.572197478039275, −6.062060421879835, −5.154387956937845, −4.505388584378325, −4.374796172764042, −3.469828443975788, −2.653040956108134, −2.211737690698459, −1.495312428222691, −0.5964078377356274,
0.5964078377356274, 1.495312428222691, 2.211737690698459, 2.653040956108134, 3.469828443975788, 4.374796172764042, 4.505388584378325, 5.154387956937845, 6.062060421879835, 6.572197478039275, 7.118944047595196, 7.814919782442171, 8.051245871883114, 8.835556730971857, 9.290708119978733, 9.668775549366504, 10.29001460372594, 11.01719976917835, 11.37615591573045, 11.91737823663184, 12.50362544334486, 12.91076493928445, 13.59102018218276, 14.18046090678456, 14.35073320452132
