| L(s) = 1 | + 2-s − 3-s + 4-s + 2.78·5-s − 6-s + 7-s + 8-s + 9-s + 2.78·10-s + 11-s − 12-s − 13-s + 14-s − 2.78·15-s + 16-s + 1.43·17-s + 18-s + 2.68·19-s + 2.78·20-s − 21-s + 22-s − 5.88·23-s − 24-s + 2.77·25-s − 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.24·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.881·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.719·15-s + 0.250·16-s + 0.348·17-s + 0.235·18-s + 0.616·19-s + 0.623·20-s − 0.218·21-s + 0.213·22-s − 1.22·23-s − 0.204·24-s + 0.554·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.817207289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.817207289\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2.78T + 5T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 - 2.68T + 19T^{2} \) |
| 23 | \( 1 + 5.88T + 23T^{2} \) |
| 29 | \( 1 + 3.89T + 29T^{2} \) |
| 31 | \( 1 - 7.96T + 31T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 - 6.97T + 47T^{2} \) |
| 53 | \( 1 - 1.26T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 6.91T + 61T^{2} \) |
| 67 | \( 1 + 1.98T + 67T^{2} \) |
| 71 | \( 1 - 5.66T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 2.46T + 89T^{2} \) |
| 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
−7.84793611743140193757647856771, −7.25368908223255420230280283596, −6.29029010493348130953619056396, −5.93129219456939121784592875818, −5.33069972678468664781888402337, −4.57617589757603541784719192131, −3.82451469635619909604287646201, −2.67072265030436922816779891948, −1.94932413410876436335128229850, −1.00206763344678882853179885200,
1.00206763344678882853179885200, 1.94932413410876436335128229850, 2.67072265030436922816779891948, 3.82451469635619909604287646201, 4.57617589757603541784719192131, 5.33069972678468664781888402337, 5.93129219456939121784592875818, 6.29029010493348130953619056396, 7.25368908223255420230280283596, 7.84793611743140193757647856771
