| L(s) = 1 | + 2-s + 2·3-s + 4-s + 5-s + 2·6-s + 3·7-s + 8-s + 9-s + 10-s − 5·11-s + 2·12-s + 3·14-s + 2·15-s + 16-s + 2·17-s + 18-s + 2·19-s + 20-s + 6·21-s − 5·22-s − 4·23-s + 2·24-s + 25-s − 4·27-s + 3·28-s − 29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.50·11-s + 0.577·12-s + 0.801·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.458·19-s + 0.223·20-s + 1.30·21-s − 1.06·22-s − 0.834·23-s + 0.408·24-s + 1/5·25-s − 0.769·27-s + 0.566·28-s − 0.185·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.83043446527026, −14.37415977203348, −14.11685764906819, −13.61835738169152, −13.07506702565298, −12.67334424041476, −12.01377850058965, −11.30615087326428, −10.99146331903150, −10.21993588445327, −9.875898412135587, −9.156352823090047, −8.377706097536387, −8.191203371438000, −7.539548298274017, −7.212169785607195, −6.223144138472771, −5.460178067973073, −5.265091092261242, −4.585645256906540, −3.775820777936000, −3.168360595585164, −2.659157308045440, −1.868040532512871, −1.587311440203644, 0,
1.587311440203644, 1.868040532512871, 2.659157308045440, 3.168360595585164, 3.775820777936000, 4.585645256906540, 5.265091092261242, 5.460178067973073, 6.223144138472771, 7.212169785607195, 7.539548298274017, 8.191203371438000, 8.377706097536387, 9.156352823090047, 9.875898412135587, 10.21993588445327, 10.99146331903150, 11.30615087326428, 12.01377850058965, 12.67334424041476, 13.07506702565298, 13.61835738169152, 14.11685764906819, 14.37415977203348, 14.83043446527026
