| L(s) = 1 | + 2-s − 3-s − 4-s − 2·5-s − 6-s − 3·8-s + 9-s − 2·10-s + 12-s − 6·13-s + 2·15-s − 16-s + 6·17-s + 18-s − 19-s + 2·20-s − 4·23-s + 3·24-s − 25-s − 6·26-s − 27-s − 2·29-s + 2·30-s + 5·32-s + 6·34-s − 36-s + 10·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s − 1.66·13-s + 0.516·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.229·19-s + 0.447·20-s − 0.834·23-s + 0.612·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.371·29-s + 0.365·30-s + 0.883·32-s + 1.02·34-s − 1/6·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54777 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5402692581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5402692581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.55830932417369, −13.98484653599343, −13.33743455822327, −12.73193762013542, −12.35473279453506, −11.95038206829991, −11.70003122934135, −10.96956291754853, −10.23385441340449, −9.808046517016742, −9.418102017555300, −8.660349936053549, −7.962433678519572, −7.535738547627834, −7.199271654522254, −6.145465006116439, −5.840692843495436, −5.251794926939652, −4.602267299444717, −4.247515476367365, −3.675493002778125, −2.958708260013087, −2.323112880156920, −1.162101435981243, −0.2610285397563646,
0.2610285397563646, 1.162101435981243, 2.323112880156920, 2.958708260013087, 3.675493002778125, 4.247515476367365, 4.602267299444717, 5.251794926939652, 5.840692843495436, 6.145465006116439, 7.199271654522254, 7.535738547627834, 7.962433678519572, 8.660349936053549, 9.418102017555300, 9.808046517016742, 10.23385441340449, 10.96956291754853, 11.70003122934135, 11.95038206829991, 12.35473279453506, 12.73193762013542, 13.33743455822327, 13.98484653599343, 14.55830932417369
