| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 3·11-s − 5·13-s − 14-s + 16-s − 6·19-s + 20-s + 3·22-s − 8·23-s − 4·25-s − 5·26-s − 28-s + 4·29-s + 6·31-s + 32-s − 35-s + 5·37-s − 6·38-s + 40-s − 4·41-s + 43-s + 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.904·11-s − 1.38·13-s − 0.267·14-s + 1/4·16-s − 1.37·19-s + 0.223·20-s + 0.639·22-s − 1.66·23-s − 4/5·25-s − 0.980·26-s − 0.188·28-s + 0.742·29-s + 1.07·31-s + 0.176·32-s − 0.169·35-s + 0.821·37-s − 0.973·38-s + 0.158·40-s − 0.624·41-s + 0.152·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.989395567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.989395567\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−14.86876223363718, −14.27108103113269, −13.94011200836572, −13.39324277483193, −12.79462773487516, −12.24169593726323, −11.93377190525211, −11.44829550603992, −10.57025515972169, −10.11458391040835, −9.710461400908999, −9.178389277065886, −8.248200870771146, −7.974353076272996, −7.069709022612900, −6.538802712132373, −6.196409871521924, −5.576543308939058, −4.800533422684612, −4.243169050775416, −3.816934905886888, −2.857081232736304, −2.260811270594709, −1.752803710610900, −0.5379424174256485,
0.5379424174256485, 1.752803710610900, 2.260811270594709, 2.857081232736304, 3.816934905886888, 4.243169050775416, 4.800533422684612, 5.576543308939058, 6.196409871521924, 6.538802712132373, 7.069709022612900, 7.974353076272996, 8.248200870771146, 9.178389277065886, 9.710461400908999, 10.11458391040835, 10.57025515972169, 11.44829550603992, 11.93377190525211, 12.24169593726323, 12.79462773487516, 13.39324277483193, 13.94011200836572, 14.27108103113269, 14.86876223363718
