| L(s) = 1 | − 1.30·2-s − 2.30·3-s − 0.302·4-s + 3·6-s + 0.697·7-s + 3·8-s + 2.30·9-s + 0.697·12-s + 5·13-s − 0.908·14-s − 3.30·16-s + 6.90·17-s − 3.00·18-s + 19-s − 1.60·21-s + 7.30·23-s − 6.90·24-s − 6.51·26-s + 1.60·27-s − 0.211·28-s − 0.908·29-s + 10.2·31-s − 1.69·32-s − 9·34-s − 0.697·36-s − 2.39·37-s − 1.30·38-s + ⋯ |
| L(s) = 1 | − 0.921·2-s − 1.32·3-s − 0.151·4-s + 1.22·6-s + 0.263·7-s + 1.06·8-s + 0.767·9-s + 0.201·12-s + 1.38·13-s − 0.242·14-s − 0.825·16-s + 1.67·17-s − 0.707·18-s + 0.229·19-s − 0.350·21-s + 1.52·23-s − 1.41·24-s − 1.27·26-s + 0.308·27-s − 0.0398·28-s − 0.168·29-s + 1.83·31-s − 0.300·32-s − 1.54·34-s − 0.116·36-s − 0.393·37-s − 0.211·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8049434396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8049434396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 3 | \( 1 + 2.30T + 3T^{2} \) |
| 7 | \( 1 - 0.697T + 7T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 7.30T + 23T^{2} \) |
| 29 | \( 1 + 0.908T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 2.39T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 1.30T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 7.90T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 2.60T + 71T^{2} \) |
| 73 | \( 1 + 7.90T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 3.51T + 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.713450936645175246796066135305, −8.055367724053810165337883082103, −7.32467887780684844506198443288, −6.43087032433346427736162408510, −5.68989340478594701629345532959, −5.02905188524262432785970364927, −4.21284388153124364065007740034, −3.07377780387802733740678545292, −1.27551975892437191760897187035, −0.834870176507958463074419368387,
0.834870176507958463074419368387, 1.27551975892437191760897187035, 3.07377780387802733740678545292, 4.21284388153124364065007740034, 5.02905188524262432785970364927, 5.68989340478594701629345532959, 6.43087032433346427736162408510, 7.32467887780684844506198443288, 8.055367724053810165337883082103, 8.713450936645175246796066135305
