| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 4·7-s − 8-s + 9-s − 10-s + 11-s + 12-s + 2·13-s − 4·14-s + 15-s + 16-s − 17-s − 18-s + 4·19-s + 20-s + 4·21-s − 22-s − 8·23-s − 24-s + 25-s − 2·26-s + 27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.872·21-s − 0.213·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.643728362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.643728362\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.029171903205937171894718305566, −7.82635899166214113271453685596, −6.89207253035368937368001811935, −6.06388305343208305858694670356, −5.32401757349352464182213629196, −4.42405377469196818805056220564, −3.63341037374635474095785279016, −2.47450888618903312270200675435, −1.80018890335959571843848326076, −1.01695705936018577697937350218,
1.01695705936018577697937350218, 1.80018890335959571843848326076, 2.47450888618903312270200675435, 3.63341037374635474095785279016, 4.42405377469196818805056220564, 5.32401757349352464182213629196, 6.06388305343208305858694670356, 6.89207253035368937368001811935, 7.82635899166214113271453685596, 8.029171903205937171894718305566
