| L(s) = 1 | − 2-s − 4-s + 2·7-s + 3·8-s − 3·9-s − 4·11-s − 2·13-s − 2·14-s − 16-s + 4·17-s + 3·18-s + 19-s + 4·22-s − 6·23-s + 2·26-s − 2·28-s − 6·29-s − 4·31-s − 5·32-s − 4·34-s + 3·36-s − 10·37-s − 38-s − 10·41-s + 2·43-s + 4·44-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s − 9-s − 1.20·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s + 0.970·17-s + 0.707·18-s + 0.229·19-s + 0.852·22-s − 1.25·23-s + 0.392·26-s − 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.883·32-s − 0.685·34-s + 1/2·36-s − 1.64·37-s − 0.162·38-s − 1.56·41-s + 0.304·43-s + 0.603·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−10.37126018310212384850887592042, −9.733312390315199950399079937400, −8.585360868393304501131631040207, −8.062630957458166067193418441089, −7.30449183870349658498149256559, −5.53012677110652044639814137101, −5.06192463633366510578493432028, −3.55403121637170592856879586722, −1.96814357717464198506340057617, 0,
1.96814357717464198506340057617, 3.55403121637170592856879586722, 5.06192463633366510578493432028, 5.53012677110652044639814137101, 7.30449183870349658498149256559, 8.062630957458166067193418441089, 8.585360868393304501131631040207, 9.733312390315199950399079937400, 10.37126018310212384850887592042
