| L(s) = 1 | + 2-s − 3-s + 4-s + 4·5-s − 6-s + 7-s + 8-s − 2·9-s + 4·10-s − 12-s + 2·13-s + 14-s − 4·15-s + 16-s − 5·17-s − 2·18-s + 7·19-s + 4·20-s − 21-s − 24-s + 11·25-s + 2·26-s + 5·27-s + 28-s + 8·29-s − 4·30-s − 2·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 1.26·10-s − 0.288·12-s + 0.554·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s − 1.21·17-s − 0.471·18-s + 1.60·19-s + 0.894·20-s − 0.218·21-s − 0.204·24-s + 11/5·25-s + 0.392·26-s + 0.962·27-s + 0.188·28-s + 1.48·29-s − 0.730·30-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.140162079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.140162079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.76273982396585846042929644753, −6.78349094051648716376911583210, −6.32919144026309965242726853065, −5.77149914675595827442226577980, −5.11451297704310861295480639903, −4.76392994502934932432893232728, −3.44796159674941176167599088898, −2.66863656315863276730725056024, −1.90351170539539027420965315976, −0.992289632248021620703998351670,
0.992289632248021620703998351670, 1.90351170539539027420965315976, 2.66863656315863276730725056024, 3.44796159674941176167599088898, 4.76392994502934932432893232728, 5.11451297704310861295480639903, 5.77149914675595827442226577980, 6.32919144026309965242726853065, 6.78349094051648716376911583210, 7.76273982396585846042929644753
