| L(s) = 1 | − 3-s + 4·7-s + 9-s + 4·11-s − 4·13-s − 2·17-s − 4·19-s − 4·21-s + 8·23-s − 5·25-s − 27-s + 8·29-s + 4·31-s − 4·33-s + 4·37-s + 4·39-s + 6·41-s + 4·43-s + 8·47-s + 9·49-s + 2·51-s + 8·53-s + 4·57-s − 12·59-s − 12·61-s + 4·63-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s − 0.485·17-s − 0.917·19-s − 0.872·21-s + 1.66·23-s − 25-s − 0.192·27-s + 1.48·29-s + 0.718·31-s − 0.696·33-s + 0.657·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 0.280·51-s + 1.09·53-s + 0.529·57-s − 1.56·59-s − 1.53·61-s + 0.503·63-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.536299908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536299908\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.55369204487954849016777034475, −9.426956093382149329706606279385, −8.649901276959931744958431496842, −7.67021342089506565111723692624, −6.85175430383558681973846914652, −5.88446326475989803483764829308, −4.70084538751769099501127599916, −4.33363432781740332926422055062, −2.46196604328535614137231641027, −1.15979350949225944879282791156,
1.15979350949225944879282791156, 2.46196604328535614137231641027, 4.33363432781740332926422055062, 4.70084538751769099501127599916, 5.88446326475989803483764829308, 6.85175430383558681973846914652, 7.67021342089506565111723692624, 8.649901276959931744958431496842, 9.426956093382149329706606279385, 10.55369204487954849016777034475
