| L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s + 4·11-s + 13-s − 14-s + 16-s + 2·17-s − 4·19-s + 2·20-s + 4·22-s + 4·23-s − 25-s + 26-s − 28-s + 2·29-s + 32-s + 2·34-s − 2·35-s − 2·37-s − 4·38-s + 2·40-s − 2·41-s + 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 1.20·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.447·20-s + 0.852·22-s + 0.834·23-s − 1/5·25-s + 0.196·26-s − 0.188·28-s + 0.371·29-s + 0.176·32-s + 0.342·34-s − 0.338·35-s − 0.328·37-s − 0.648·38-s + 0.316·40-s − 0.312·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.315613366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.315613366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.344633920829121985006795507809, −8.793062426761640380192286442759, −7.60950019354364822141375551305, −6.62410620605336828325420319307, −6.18971912035243321530580658857, −5.36734378010452113216798273238, −4.32571873646081315893153040604, −3.48844535790283312124632987251, −2.39519789676916156425539342752, −1.29507569141715195434836244797,
1.29507569141715195434836244797, 2.39519789676916156425539342752, 3.48844535790283312124632987251, 4.32571873646081315893153040604, 5.36734378010452113216798273238, 6.18971912035243321530580658857, 6.62410620605336828325420319307, 7.60950019354364822141375551305, 8.793062426761640380192286442759, 9.344633920829121985006795507809
