| L(s) = 1 | + 5·2-s − 9·3-s − 7·4-s − 94·5-s − 45·6-s − 195·8-s + 81·9-s − 470·10-s + 52·11-s + 63·12-s + 770·13-s + 846·15-s − 751·16-s + 2.02e3·17-s + 405·18-s − 1.73e3·19-s + 658·20-s + 260·22-s − 576·23-s + 1.75e3·24-s + 5.71e3·25-s + 3.85e3·26-s − 729·27-s + 5.51e3·29-s + 4.23e3·30-s − 6.33e3·31-s + 2.48e3·32-s + ⋯ |
| L(s) = 1 | + 0.883·2-s − 0.577·3-s − 0.218·4-s − 1.68·5-s − 0.510·6-s − 1.07·8-s + 1/3·9-s − 1.48·10-s + 0.129·11-s + 0.126·12-s + 1.26·13-s + 0.970·15-s − 0.733·16-s + 1.69·17-s + 0.294·18-s − 1.10·19-s + 0.367·20-s + 0.114·22-s − 0.227·23-s + 0.621·24-s + 1.82·25-s + 1.11·26-s − 0.192·27-s + 1.21·29-s + 0.858·30-s − 1.18·31-s + 0.428·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.296948025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.296948025\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5 T + p^{5} T^{2} \) |
| 5 | \( 1 + 94 T + p^{5} T^{2} \) |
| 11 | \( 1 - 52 T + p^{5} T^{2} \) |
| 13 | \( 1 - 770 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2022 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1732 T + p^{5} T^{2} \) |
| 23 | \( 1 + 576 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5518 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6336 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7338 T + p^{5} T^{2} \) |
| 41 | \( 1 - 3262 T + p^{5} T^{2} \) |
| 43 | \( 1 - 5420 T + p^{5} T^{2} \) |
| 47 | \( 1 + 864 T + p^{5} T^{2} \) |
| 53 | \( 1 - 4182 T + p^{5} T^{2} \) |
| 59 | \( 1 - 11220 T + p^{5} T^{2} \) |
| 61 | \( 1 - 45602 T + p^{5} T^{2} \) |
| 67 | \( 1 - 1396 T + p^{5} T^{2} \) |
| 71 | \( 1 - 18720 T + p^{5} T^{2} \) |
| 73 | \( 1 + 46362 T + p^{5} T^{2} \) |
| 79 | \( 1 - 97424 T + p^{5} T^{2} \) |
| 83 | \( 1 - 81228 T + p^{5} T^{2} \) |
| 89 | \( 1 - 3182 T + p^{5} T^{2} \) |
| 97 | \( 1 + 4914 T + p^{5} T^{2} \) |
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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
−12.21485632817839015510265768756, −11.49303513537901796784712991864, −10.46150239206066121644864554091, −8.809672259348629098470991054065, −7.897917672598722945436767868847, −6.52273569613945244008155070798, −5.32641712009082211318677289765, −4.10255099426041650732180778876, −3.46774098137012815729971755449, −0.68245857458635360714142791644,
0.68245857458635360714142791644, 3.46774098137012815729971755449, 4.10255099426041650732180778876, 5.32641712009082211318677289765, 6.52273569613945244008155070798, 7.897917672598722945436767868847, 8.809672259348629098470991054065, 10.46150239206066121644864554091, 11.49303513537901796784712991864, 12.21485632817839015510265768756
