| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 3·9-s + 2·11-s + 6·13-s − 4·14-s + 16-s + 2·17-s + 3·18-s − 2·19-s − 2·22-s + 23-s − 5·25-s − 6·26-s + 4·28-s − 29-s − 32-s − 2·34-s − 3·36-s + 4·37-s + 2·38-s − 2·41-s + 10·43-s + 2·44-s − 46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 9-s + 0.603·11-s + 1.66·13-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.458·19-s − 0.426·22-s + 0.208·23-s − 25-s − 1.17·26-s + 0.755·28-s − 0.185·29-s − 0.176·32-s − 0.342·34-s − 1/2·36-s + 0.657·37-s + 0.324·38-s − 0.312·41-s + 1.52·43-s + 0.301·44-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.482570899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.482570899\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−9.362840693737650936052402691335, −8.737259013171114215932090957122, −8.136313333585105525919773653064, −7.51266302567345812935898598967, −6.18882132913877394849460983797, −5.72842264193310018375453685876, −4.47449759907179690290855698049, −3.44148842054825095067948308700, −2.07455635641233909533092691275, −1.06802436445112848630233397781,
1.06802436445112848630233397781, 2.07455635641233909533092691275, 3.44148842054825095067948308700, 4.47449759907179690290855698049, 5.72842264193310018375453685876, 6.18882132913877394849460983797, 7.51266302567345812935898598967, 8.136313333585105525919773653064, 8.737259013171114215932090957122, 9.362840693737650936052402691335
