| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 11-s − 7·13-s + 16-s − 4·17-s − 19-s + 20-s − 22-s − 23-s + 25-s + 7·26-s + 8·29-s − 6·31-s − 32-s + 4·34-s − 3·37-s + 38-s − 40-s + 9·41-s − 4·43-s + 44-s + 46-s − 3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 0.301·11-s − 1.94·13-s + 1/4·16-s − 0.970·17-s − 0.229·19-s + 0.223·20-s − 0.213·22-s − 0.208·23-s + 1/5·25-s + 1.37·26-s + 1.48·29-s − 1.07·31-s − 0.176·32-s + 0.685·34-s − 0.493·37-s + 0.162·38-s − 0.158·40-s + 1.40·41-s − 0.609·43-s + 0.150·44-s + 0.147·46-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.124210926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.124210926\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.396627832349766845018692016394, −7.69234677254200114708354751195, −6.84239878685049028219148372324, −6.52058887037352170698991748319, −5.36057786099946828982465209074, −4.78420387569213441370931648811, −3.72650469584049527071254512684, −2.48128176333592537039367528987, −2.09100988764213577802437323995, −0.63547758295212514566981031432,
0.63547758295212514566981031432, 2.09100988764213577802437323995, 2.48128176333592537039367528987, 3.72650469584049527071254512684, 4.78420387569213441370931648811, 5.36057786099946828982465209074, 6.52058887037352170698991748319, 6.84239878685049028219148372324, 7.69234677254200114708354751195, 8.396627832349766845018692016394
