| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s − 2·9-s − 10-s − 6·11-s + 12-s + 13-s − 15-s + 16-s − 17-s − 2·18-s + 19-s − 20-s − 6·22-s + 4·23-s + 24-s − 4·25-s + 26-s − 5·27-s − 5·29-s − 30-s + 2·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 1.80·11-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.471·18-s + 0.229·19-s − 0.223·20-s − 1.27·22-s + 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.196·26-s − 0.962·27-s − 0.928·29-s − 0.182·30-s + 0.359·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.272355323316919393921254955505, −7.75774280543920204347405913458, −7.07313263463553623662071127160, −5.91759626806284257380951582754, −5.34039928905564076506604369821, −4.51450522674191385037435684251, −3.41704014115911379391069471129, −2.90445429715702425748217165554, −1.93519265645498840362182949378, 0,
1.93519265645498840362182949378, 2.90445429715702425748217165554, 3.41704014115911379391069471129, 4.51450522674191385037435684251, 5.34039928905564076506604369821, 5.91759626806284257380951582754, 7.07313263463553623662071127160, 7.75774280543920204347405913458, 8.272355323316919393921254955505
