| L(s) = 1 | + 2.60·2-s − 3-s + 4.76·4-s + 5-s − 2.60·6-s + 3.60·7-s + 7.20·8-s + 9-s + 2.60·10-s − 5.20·11-s − 4.76·12-s + 9.37·14-s − 15-s + 9.20·16-s − 2.93·17-s + 2.60·18-s + 6.76·19-s + 4.76·20-s − 3.60·21-s − 13.5·22-s + 5.53·23-s − 7.20·24-s + 25-s − 27-s + 17.1·28-s + 1.83·29-s − 2.60·30-s + ⋯ |
| L(s) = 1 | + 1.83·2-s − 0.577·3-s + 2.38·4-s + 0.447·5-s − 1.06·6-s + 1.36·7-s + 2.54·8-s + 0.333·9-s + 0.822·10-s − 1.56·11-s − 1.37·12-s + 2.50·14-s − 0.258·15-s + 2.30·16-s − 0.712·17-s + 0.613·18-s + 1.55·19-s + 1.06·20-s − 0.785·21-s − 2.88·22-s + 1.15·23-s − 1.47·24-s + 0.200·25-s − 0.192·27-s + 3.24·28-s + 0.340·29-s − 0.474·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.901922891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.901922891\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 + 5.20T + 11T^{2} \) |
| 17 | \( 1 + 2.93T + 17T^{2} \) |
| 19 | \( 1 - 6.76T + 19T^{2} \) |
| 23 | \( 1 - 5.53T + 23T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 31 | \( 1 - 4.10T + 31T^{2} \) |
| 37 | \( 1 + 3.53T + 37T^{2} \) |
| 41 | \( 1 + 5.37T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 - 3.80T + 47T^{2} \) |
| 53 | \( 1 + 5.20T + 53T^{2} \) |
| 59 | \( 1 + 7.37T + 59T^{2} \) |
| 61 | \( 1 + 3.43T + 61T^{2} \) |
| 67 | \( 1 - 3.50T + 67T^{2} \) |
| 71 | \( 1 + 9.70T + 71T^{2} \) |
| 73 | \( 1 + 0.805T + 73T^{2} \) |
| 79 | \( 1 + 4.10T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 9.83T + 89T^{2} \) |
| 97 | \( 1 + 5.57T + 97T^{2} \) |
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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
−8.752360521794356883089690204386, −7.67001967339431837056579034132, −7.23828357222469569841866153965, −6.22852068467580380019077221423, −5.44869910711496184286412109401, −4.94906115153582998029878879676, −4.59790466331063369792514803537, −3.22764290651301528845936300125, −2.44506953035160570273660974567, −1.39857396900715405153260297307,
1.39857396900715405153260297307, 2.44506953035160570273660974567, 3.22764290651301528845936300125, 4.59790466331063369792514803537, 4.94906115153582998029878879676, 5.44869910711496184286412109401, 6.22852068467580380019077221423, 7.23828357222469569841866153965, 7.67001967339431837056579034132, 8.752360521794356883089690204386
