| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s + 3·8-s + 9-s + 10-s − 4·11-s − 12-s + 2·13-s − 15-s − 16-s − 2·17-s − 18-s − 4·19-s + 20-s + 4·22-s + 3·24-s + 25-s − 2·26-s + 27-s − 2·29-s + 30-s − 5·32-s − 4·33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.883·32-s − 0.696·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.948652838212898735141314289502, −8.838960281218345298135626667274, −8.413264856684473334036504393340, −7.68993943182315499613922198544, −6.74126848758315603293550810292, −5.28294324247232318062401992920, −4.36459226831434420870816413097, −3.30593125964790138739196334719, −1.84339419921478037654642459241, 0,
1.84339419921478037654642459241, 3.30593125964790138739196334719, 4.36459226831434420870816413097, 5.28294324247232318062401992920, 6.74126848758315603293550810292, 7.68993943182315499613922198544, 8.413264856684473334036504393340, 8.838960281218345298135626667274, 9.948652838212898735141314289502
