| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 5·5-s + 6·6-s − 24·7-s + 8·8-s + 9·9-s − 10·10-s + 32·11-s + 12·12-s + 2·13-s − 48·14-s − 15·15-s + 16·16-s + 106·17-s + 18·18-s − 19·19-s − 20·20-s − 72·21-s + 64·22-s + 152·23-s + 24·24-s + 25·25-s + 4·26-s + 27·27-s − 96·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.29·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.877·11-s + 0.288·12-s + 0.0426·13-s − 0.916·14-s − 0.258·15-s + 1/4·16-s + 1.51·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.748·21-s + 0.620·22-s + 1.37·23-s + 0.204·24-s + 1/5·25-s + 0.0301·26-s + 0.192·27-s − 0.647·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.483428610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.483428610\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 19 | \( 1 + p T \) |
| good | 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 32 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 106 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 - 52 T + p^{3} T^{2} \) |
| 37 | \( 1 - 306 T + p^{3} T^{2} \) |
| 41 | \( 1 - 62 T + p^{3} T^{2} \) |
| 43 | \( 1 + 268 T + p^{3} T^{2} \) |
| 47 | \( 1 - 456 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 300 T + p^{3} T^{2} \) |
| 61 | \( 1 - 502 T + p^{3} T^{2} \) |
| 67 | \( 1 + 644 T + p^{3} T^{2} \) |
| 71 | \( 1 + 608 T + p^{3} T^{2} \) |
| 73 | \( 1 + 198 T + p^{3} T^{2} \) |
| 79 | \( 1 - 260 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1248 T + p^{3} T^{2} \) |
| 89 | \( 1 - 110 T + p^{3} T^{2} \) |
| 97 | \( 1 + 574 T + p^{3} T^{2} \) |
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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
−10.23623038605908190808232948239, −9.521469118071580044830689308543, −8.583885420988029923586980335297, −7.45928328487433348085019486392, −6.70794338109015669392044513580, −5.79830765616823864978024595215, −4.45241796664178277240704763942, −3.47959467311618260848227616346, −2.84171668861277630555181671360, −1.04100769578237151327975940888,
1.04100769578237151327975940888, 2.84171668861277630555181671360, 3.47959467311618260848227616346, 4.45241796664178277240704763942, 5.79830765616823864978024595215, 6.70794338109015669392044513580, 7.45928328487433348085019486392, 8.583885420988029923586980335297, 9.521469118071580044830689308543, 10.23623038605908190808232948239
