| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 4.59·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s − 9.18·10-s − 45.1·11-s + 12·12-s + 13·13-s − 14·14-s + 13.7·15-s + 16·16-s − 93.5·17-s − 18·18-s − 35.7·19-s + 18.3·20-s + 21·21-s + 90.3·22-s − 79.7·23-s − 24·24-s − 103.·25-s − 26·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.410·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.290·10-s − 1.23·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.237·15-s + 0.250·16-s − 1.33·17-s − 0.235·18-s − 0.432·19-s + 0.205·20-s + 0.218·21-s + 0.875·22-s − 0.723·23-s − 0.204·24-s − 0.831·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 4.59T + 125T^{2} \) |
| 11 | \( 1 + 45.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 93.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 79.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 13.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 71.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 42.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 53.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 4.65T + 7.95e4T^{2} \) |
| 47 | \( 1 - 21.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 258.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 336T + 2.05e5T^{2} \) |
| 61 | \( 1 - 230T + 2.26e5T^{2} \) |
| 67 | \( 1 + 408.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 92.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 621.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 350.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 336.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 854.T + 9.12e5T^{2} \) |
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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
−9.927006288622326114750115028881, −8.980049234937646047137676256524, −8.264759021117413018040745060385, −7.51876184573162009320945293309, −6.44873346974281539409842078203, −5.36579668752903462472372727395, −4.08370138530358047333084710297, −2.60625706564551040449180524064, −1.80173158755395749595614850502, 0,
1.80173158755395749595614850502, 2.60625706564551040449180524064, 4.08370138530358047333084710297, 5.36579668752903462472372727395, 6.44873346974281539409842078203, 7.51876184573162009320945293309, 8.264759021117413018040745060385, 8.980049234937646047137676256524, 9.927006288622326114750115028881
