| L(s) = 1 | − 0.874·3-s − 2.23·9-s + 3.47·11-s + 2.28·13-s − 1.74·17-s − 0.333·19-s + 5.47·23-s + 4.57·27-s − 4.23·29-s − 1.20·31-s − 3.03·33-s − 0.236·37-s − 2·39-s + 1.95·41-s + 8.23·43-s − 7.73·47-s + 1.52·51-s − 1.70·53-s + 0.291·57-s − 5.11·59-s − 14.6·61-s + 3.94·67-s − 4.78·69-s + 3.29·71-s + 14.6·73-s − 2.52·79-s + 2.70·81-s + ⋯ |
| L(s) = 1 | − 0.504·3-s − 0.745·9-s + 1.04·11-s + 0.634·13-s − 0.423·17-s − 0.0765·19-s + 1.14·23-s + 0.880·27-s − 0.786·29-s − 0.216·31-s − 0.528·33-s − 0.0388·37-s − 0.320·39-s + 0.305·41-s + 1.25·43-s − 1.12·47-s + 0.213·51-s − 0.234·53-s + 0.0386·57-s − 0.666·59-s − 1.86·61-s + 0.481·67-s − 0.575·69-s + 0.390·71-s + 1.70·73-s − 0.284·79-s + 0.300·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.510064545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.510064545\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.874T + 3T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 + 0.333T + 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 + 1.20T + 31T^{2} \) |
| 37 | \( 1 + 0.236T + 37T^{2} \) |
| 41 | \( 1 - 1.95T + 41T^{2} \) |
| 43 | \( 1 - 8.23T + 43T^{2} \) |
| 47 | \( 1 + 7.73T + 47T^{2} \) |
| 53 | \( 1 + 1.70T + 53T^{2} \) |
| 59 | \( 1 + 5.11T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 3.94T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 2.52T + 79T^{2} \) |
| 83 | \( 1 - 9.02T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−8.311652552449682525809245862996, −7.50790686304150118934254663153, −6.59377569708289552672688706182, −6.20425275236286522213559712120, −5.39558547170163201937129346998, −4.62822159024285708917792211729, −3.73805032795989197811875815239, −2.96573568440868785956777572799, −1.78964527129714277750024450362, −0.70399133812821928059399076447,
0.70399133812821928059399076447, 1.78964527129714277750024450362, 2.96573568440868785956777572799, 3.73805032795989197811875815239, 4.62822159024285708917792211729, 5.39558547170163201937129346998, 6.20425275236286522213559712120, 6.59377569708289552672688706182, 7.50790686304150118934254663153, 8.311652552449682525809245862996
