| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 7-s − 3·8-s + 9-s + 4·11-s + 12-s + 2·13-s + 14-s − 16-s + 6·17-s + 18-s + 4·19-s − 21-s + 4·22-s + 3·24-s + 2·26-s − 27-s − 28-s − 2·29-s + 5·32-s − 4·33-s + 6·34-s − 36-s − 6·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.218·21-s + 0.852·22-s + 0.612·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.883·32-s − 0.696·33-s + 1.02·34-s − 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.613959539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.613959539\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−11.11201786491621495190447404117, −9.925735637620119905720933688370, −9.207296367580559757990715462054, −8.195189929730273740680390755697, −7.03072366414782310604425087892, −5.91493242676799986091032287241, −5.29867067368365038443059808047, −4.17885429573697350339696872585, −3.34454169743649929013210973301, −1.19446009883044712799493276453,
1.19446009883044712799493276453, 3.34454169743649929013210973301, 4.17885429573697350339696872585, 5.29867067368365038443059808047, 5.91493242676799986091032287241, 7.03072366414782310604425087892, 8.195189929730273740680390755697, 9.207296367580559757990715462054, 9.925735637620119905720933688370, 11.11201786491621495190447404117
