| L(s) = 1 | + 3·3-s + 5-s + 6·9-s − 11-s − 2·13-s + 3·15-s + 3·17-s + 5·19-s + 3·23-s − 4·25-s + 9·27-s + 6·29-s + 31-s − 3·33-s + 5·37-s − 6·39-s − 10·41-s − 4·43-s + 6·45-s − 47-s + 9·51-s + 9·53-s − 55-s + 15·57-s + 3·59-s − 3·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s − 0.301·11-s − 0.554·13-s + 0.774·15-s + 0.727·17-s + 1.14·19-s + 0.625·23-s − 4/5·25-s + 1.73·27-s + 1.11·29-s + 0.179·31-s − 0.522·33-s + 0.821·37-s − 0.960·39-s − 1.56·41-s − 0.609·43-s + 0.894·45-s − 0.145·47-s + 1.26·51-s + 1.23·53-s − 0.134·55-s + 1.98·57-s + 0.390·59-s − 0.384·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.039410017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.039410017\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−8.613387530453077204772370730115, −8.031996104849329823930918913127, −7.41222969775341712005479565068, −6.69932322919898823116211066736, −5.52190697314800540673323359415, −4.74391206643302649394539653497, −3.67640048216274307516056199890, −2.98885024194690194695850849492, −2.28746912531373009454400424430, −1.23644851964910009262807684617,
1.23644851964910009262807684617, 2.28746912531373009454400424430, 2.98885024194690194695850849492, 3.67640048216274307516056199890, 4.74391206643302649394539653497, 5.52190697314800540673323359415, 6.69932322919898823116211066736, 7.41222969775341712005479565068, 8.031996104849329823930918913127, 8.613387530453077204772370730115
