| L(s) = 1 | − 1.59·3-s + 2.24·5-s + 2.30·7-s − 0.445·9-s + 4.16·11-s − 3.59·15-s − 0.554·17-s − 7.75·19-s − 3.69·21-s + 2.48·23-s + 0.0489·25-s + 5.50·27-s − 5.96·29-s − 2.94·31-s − 6.65·33-s + 5.19·35-s − 4.18·37-s − 11.5·41-s − 0.648·43-s − 0.999·45-s − 5.93·47-s − 1.66·49-s + 0.887·51-s + 1.40·53-s + 9.35·55-s + 12.3·57-s − 10.4·59-s + ⋯ |
| L(s) = 1 | − 0.922·3-s + 1.00·5-s + 0.873·7-s − 0.148·9-s + 1.25·11-s − 0.927·15-s − 0.134·17-s − 1.77·19-s − 0.805·21-s + 0.518·23-s + 0.00978·25-s + 1.05·27-s − 1.10·29-s − 0.528·31-s − 1.15·33-s + 0.877·35-s − 0.688·37-s − 1.81·41-s − 0.0989·43-s − 0.149·45-s − 0.865·47-s − 0.237·49-s + 0.124·51-s + 0.193·53-s + 1.26·55-s + 1.64·57-s − 1.36·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.59T + 3T^{2} \) |
| 5 | \( 1 - 2.24T + 5T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 - 4.16T + 11T^{2} \) |
| 17 | \( 1 + 0.554T + 17T^{2} \) |
| 19 | \( 1 + 7.75T + 19T^{2} \) |
| 23 | \( 1 - 2.48T + 23T^{2} \) |
| 29 | \( 1 + 5.96T + 29T^{2} \) |
| 31 | \( 1 + 2.94T + 31T^{2} \) |
| 37 | \( 1 + 4.18T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 0.648T + 43T^{2} \) |
| 47 | \( 1 + 5.93T + 47T^{2} \) |
| 53 | \( 1 - 1.40T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 2.59T + 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 - 8.92T + 83T^{2} \) |
| 89 | \( 1 - 5.47T + 89T^{2} \) |
| 97 | \( 1 + 8.18T + 97T^{2} \) |
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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
−7.80884595679647160670663038480, −6.74026299517629262271318027675, −6.38115573517461707805027622576, −5.69271070709273428701157891341, −5.00911947043402216521099022452, −4.35048105569482034251034018175, −3.31739178582970529223305062862, −1.96327777650498256327322095937, −1.53262116854449723263160816685, 0,
1.53262116854449723263160816685, 1.96327777650498256327322095937, 3.31739178582970529223305062862, 4.35048105569482034251034018175, 5.00911947043402216521099022452, 5.69271070709273428701157891341, 6.38115573517461707805027622576, 6.74026299517629262271318027675, 7.80884595679647160670663038480
