| L(s) = 1 | + 3-s − 2·9-s − 3·11-s + 7·13-s − 5·17-s − 4·19-s − 4·23-s − 5·27-s + 5·29-s − 2·31-s − 3·33-s + 7·39-s − 6·41-s − 8·43-s − 9·47-s − 5·51-s − 4·57-s − 12·59-s − 6·61-s − 16·67-s − 4·69-s + 4·71-s − 14·73-s + 9·79-s + 81-s − 8·83-s + 5·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.904·11-s + 1.94·13-s − 1.21·17-s − 0.917·19-s − 0.834·23-s − 0.962·27-s + 0.928·29-s − 0.359·31-s − 0.522·33-s + 1.12·39-s − 0.937·41-s − 1.21·43-s − 1.31·47-s − 0.700·51-s − 0.529·57-s − 1.56·59-s − 0.768·61-s − 1.95·67-s − 0.481·69-s + 0.474·71-s − 1.63·73-s + 1.01·79-s + 1/9·81-s − 0.878·83-s + 0.536·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8879340420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8879340420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.76911313361673, −13.62326533134665, −13.16653169382034, −12.75099036172529, −11.89049866365017, −11.53153592395431, −10.97527712631393, −10.48433225559440, −10.21063555206828, −9.216403354241674, −8.909157562720408, −8.380849600104089, −8.142237977783899, −7.558022331302455, −6.594857268963011, −6.335034327106272, −5.848658231683870, −5.096350654474523, −4.474236557810685, −3.916301991397829, −3.174865401276885, −2.879123058470611, −1.919936929904904, −1.588910702539144, −0.2757621890436199,
0.2757621890436199, 1.588910702539144, 1.919936929904904, 2.879123058470611, 3.174865401276885, 3.916301991397829, 4.474236557810685, 5.096350654474523, 5.848658231683870, 6.335034327106272, 6.594857268963011, 7.558022331302455, 8.142237977783899, 8.380849600104089, 8.909157562720408, 9.216403354241674, 10.21063555206828, 10.48433225559440, 10.97527712631393, 11.53153592395431, 11.89049866365017, 12.75099036172529, 13.16653169382034, 13.62326533134665, 13.76911313361673
