| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s + 2·7-s + 8-s − 2·9-s + 3·10-s − 3·11-s + 12-s + 2·13-s + 2·14-s + 3·15-s + 16-s − 2·18-s + 19-s + 3·20-s + 2·21-s − 3·22-s + 23-s + 24-s + 4·25-s + 2·26-s − 5·27-s + 2·28-s − 3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s − 0.904·11-s + 0.288·12-s + 0.554·13-s + 0.534·14-s + 0.774·15-s + 1/4·16-s − 0.471·18-s + 0.229·19-s + 0.670·20-s + 0.436·21-s − 0.639·22-s + 0.208·23-s + 0.204·24-s + 4/5·25-s + 0.392·26-s − 0.962·27-s + 0.377·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 874 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 874 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.543424156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.543424156\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−10.21643769358178065968167185911, −9.288450593505419738882905923558, −8.451290385585527310238856174142, −7.65428284872921883138219868719, −6.49300212845425635180999211709, −5.52472561366771168776296847905, −5.12646721342553207977876634965, −3.64342042429842783123261249929, −2.57151933895629597492403624282, −1.74726461685773298384380211741,
1.74726461685773298384380211741, 2.57151933895629597492403624282, 3.64342042429842783123261249929, 5.12646721342553207977876634965, 5.52472561366771168776296847905, 6.49300212845425635180999211709, 7.65428284872921883138219868719, 8.451290385585527310238856174142, 9.288450593505419738882905923558, 10.21643769358178065968167185911
