| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 2·5-s − 2·6-s + 9-s + 4·10-s − 2·11-s − 2·12-s − 13-s − 2·15-s − 4·16-s + 2·18-s − 19-s + 4·20-s − 4·22-s − 25-s − 2·26-s − 27-s + 4·29-s − 4·30-s − 9·31-s − 8·32-s + 2·33-s + 2·36-s + 3·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.894·5-s − 0.816·6-s + 1/3·9-s + 1.26·10-s − 0.603·11-s − 0.577·12-s − 0.277·13-s − 0.516·15-s − 16-s + 0.471·18-s − 0.229·19-s + 0.894·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.742·29-s − 0.730·30-s − 1.61·31-s − 1.41·32-s + 0.348·33-s + 1/3·36-s + 0.493·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.920053745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.920053745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.05041447442840207720378708897, −12.40866641446903147774840263019, −11.30162828605416851531895428330, −10.28136035144373451283122510308, −9.092563983560312555964020990816, −7.29485403540511127428179965078, −6.01621490934601452163404663718, −5.40855511243246313730427714163, −4.18739539622885934722496613081, −2.47062022887811000287520417928,
2.47062022887811000287520417928, 4.18739539622885934722496613081, 5.40855511243246313730427714163, 6.01621490934601452163404663718, 7.29485403540511127428179965078, 9.092563983560312555964020990816, 10.28136035144373451283122510308, 11.30162828605416851531895428330, 12.40866641446903147774840263019, 13.05041447442840207720378708897
