| 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 − 6·17-s + 18-s + 4·19-s − 20-s + 4·22-s − 23-s + 3·24-s + 25-s − 2·26-s − 27-s + 29-s − 30-s + 4·31-s + 5·32-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 − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.852·22-s − 0.208·23-s + 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.185·29-s − 0.182·30-s + 0.718·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−17.17718612691170, −16.30636475088869, −15.73588379599894, −15.09616656410795, −14.50618449612901, −14.01570979062440, −13.46538500862401, −13.00211296240947, −12.27194063929149, −11.80240693551841, −11.37018780481208, −10.49094406420653, −9.768763226507564, −9.338643346579857, −8.768821891142581, −7.992315866950088, −6.946005974467450, −6.489899452271880, −5.962355840346338, −5.032412579610391, −4.749854360805160, −3.949153321139294, −3.229960703715178, −2.218764488273508, −1.180916365052916, 0,
1.180916365052916, 2.218764488273508, 3.229960703715178, 3.949153321139294, 4.749854360805160, 5.032412579610391, 5.962355840346338, 6.489899452271880, 6.946005974467450, 7.992315866950088, 8.768821891142581, 9.338643346579857, 9.768763226507564, 10.49094406420653, 11.37018780481208, 11.80240693551841, 12.27194063929149, 13.00211296240947, 13.46538500862401, 14.01570979062440, 14.50618449612901, 15.09616656410795, 15.73588379599894, 16.30636475088869, 17.17718612691170
