| L(s) = 1 | + i·2-s + 2.34i·3-s − 4-s − 2.34·6-s − 3.83i·7-s − i·8-s − 2.48·9-s + 3.19·11-s − 2.34i·12-s − 1.14i·13-s + 3.83·14-s + 16-s + 0.803i·17-s − 2.48i·18-s + 2.34·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 1.35i·3-s − 0.5·4-s − 0.956·6-s − 1.44i·7-s − 0.353i·8-s − 0.829·9-s + 0.963·11-s − 0.676i·12-s − 0.317i·13-s + 1.02·14-s + 0.250·16-s + 0.194i·17-s − 0.586i·18-s + 0.537·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.669958388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.669958388\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.34iT - 3T^{2} \) |
| 7 | \( 1 + 3.83iT - 7T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 13 | \( 1 + 1.14iT - 13T^{2} \) |
| 17 | \( 1 - 0.803iT - 17T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 23 | \( 1 - 8.12iT - 23T^{2} \) |
| 31 | \( 1 - 1.90T + 31T^{2} \) |
| 37 | \( 1 - 7.63iT - 37T^{2} \) |
| 41 | \( 1 - 8.63T + 41T^{2} \) |
| 43 | \( 1 + 7.66iT - 43T^{2} \) |
| 47 | \( 1 - 11.4iT - 47T^{2} \) |
| 53 | \( 1 - 0.560iT - 53T^{2} \) |
| 59 | \( 1 + 2.36T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 5.12iT - 67T^{2} \) |
| 71 | \( 1 - 8.12T + 71T^{2} \) |
| 73 | \( 1 + 5.71iT - 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 3.09iT - 83T^{2} \) |
| 89 | \( 1 - 3.32T + 89T^{2} \) |
| 97 | \( 1 - 11.5iT - 97T^{2} \) |
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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
−9.536985833939634850437856359605, −9.340343168049685012798971203618, −8.082906184210834300022284487000, −7.39323049263240771129071739424, −6.55288882369241774600567765463, −5.52801415393760302073098110713, −4.66807970390785576666261146898, −3.91777118010830431631033251719, −3.39832252943560341707242469156, −1.12722578318187148976148965090,
0.853153442287446562542221416101, 2.07318508979689700033427751340, 2.61555166663799235648672990572, 3.97116776640865590787267829501, 5.16377279002638522520865454163, 6.12414606854524912966278591941, 6.71662946293510979331420534642, 7.74049873953026958676343531174, 8.612686480759460509980583655270, 9.102010881575976026410622867770
