| L(s) = 1 | − 2·3-s + 2·5-s + 9-s + 4·11-s − 2·13-s − 4·15-s + 2·17-s − 2·19-s + 2·23-s − 25-s + 4·27-s + 2·29-s − 2·31-s − 8·33-s + 10·37-s + 4·39-s + 2·41-s + 4·43-s + 2·45-s + 12·47-s − 7·49-s − 4·51-s − 53-s + 8·55-s + 4·57-s + 12·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 1.03·15-s + 0.485·17-s − 0.458·19-s + 0.417·23-s − 1/5·25-s + 0.769·27-s + 0.371·29-s − 0.359·31-s − 1.39·33-s + 1.64·37-s + 0.640·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.75·47-s − 49-s − 0.560·51-s − 0.137·53-s + 1.07·55-s + 0.529·57-s + 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.250182427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.250182427\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−10.17212308539052766917914385987, −9.541401430033653215631219019208, −8.677719952955944417688132044799, −7.39146525600322915154301384752, −6.44170829906824352447977127256, −5.90660064231056267066680825192, −5.07004079710150650407222317812, −4.00859464045490590884126649588, −2.43540719020094830804422818714, −1.00050848224802759305073576672,
1.00050848224802759305073576672, 2.43540719020094830804422818714, 4.00859464045490590884126649588, 5.07004079710150650407222317812, 5.90660064231056267066680825192, 6.44170829906824352447977127256, 7.39146525600322915154301384752, 8.677719952955944417688132044799, 9.541401430033653215631219019208, 10.17212308539052766917914385987
