| L(s) = 1 | + 2·2-s + 7·3-s + 4·4-s − 18·5-s + 14·6-s + 30·7-s + 8·8-s + 22·9-s − 36·10-s + 6·11-s + 28·12-s + 79·13-s + 60·14-s − 126·15-s + 16·16-s − 102·17-s + 44·18-s + 36·19-s − 72·20-s + 210·21-s + 12·22-s + 56·24-s + 199·25-s + 158·26-s − 35·27-s + 120·28-s + 33·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·3-s + 1/2·4-s − 1.60·5-s + 0.952·6-s + 1.61·7-s + 0.353·8-s + 0.814·9-s − 1.13·10-s + 0.164·11-s + 0.673·12-s + 1.68·13-s + 1.14·14-s − 2.16·15-s + 1/4·16-s − 1.45·17-s + 0.576·18-s + 0.434·19-s − 0.804·20-s + 2.18·21-s + 0.116·22-s + 0.476·24-s + 1.59·25-s + 1.19·26-s − 0.249·27-s + 0.809·28-s + 0.211·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.399947165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.399947165\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 7 | \( 1 - 30 T + p^{3} T^{2} \) |
| 11 | \( 1 - 6 T + p^{3} T^{2} \) |
| 13 | \( 1 - 79 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 36 T + p^{3} T^{2} \) |
| 29 | \( 1 - 33 T + p^{3} T^{2} \) |
| 31 | \( 1 - 43 T + p^{3} T^{2} \) |
| 37 | \( 1 - 354 T + p^{3} T^{2} \) |
| 41 | \( 1 - 375 T + p^{3} T^{2} \) |
| 43 | \( 1 + 96 T + p^{3} T^{2} \) |
| 47 | \( 1 + 129 T + p^{3} T^{2} \) |
| 53 | \( 1 - 300 T + p^{3} T^{2} \) |
| 59 | \( 1 + 324 T + p^{3} T^{2} \) |
| 61 | \( 1 + 120 T + p^{3} T^{2} \) |
| 67 | \( 1 + 582 T + p^{3} T^{2} \) |
| 71 | \( 1 - 147 T + p^{3} T^{2} \) |
| 73 | \( 1 - 637 T + p^{3} T^{2} \) |
| 79 | \( 1 - 468 T + p^{3} T^{2} \) |
| 83 | \( 1 + 978 T + p^{3} T^{2} \) |
| 89 | \( 1 - 252 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1170 T + p^{3} T^{2} \) |
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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
−9.065422902787766191681517987224, −8.484576168536895042274991860778, −7.931888574464164324866156852100, −7.37377843724436038077355207378, −6.15061688081245703993648519691, −4.68050247083329749665490942877, −4.16166956141698986518921446802, −3.44127254271915338509174205935, −2.33513053317751965166494014675, −1.12818538864848404845691384922,
1.12818538864848404845691384922, 2.33513053317751965166494014675, 3.44127254271915338509174205935, 4.16166956141698986518921446802, 4.68050247083329749665490942877, 6.15061688081245703993648519691, 7.37377843724436038077355207378, 7.931888574464164324866156852100, 8.484576168536895042274991860778, 9.065422902787766191681517987224
