| L(s) = 1 | + 3-s − 7-s + 9-s + 11-s − 6·13-s + 7·19-s − 21-s + 23-s − 5·25-s + 27-s − 10·29-s − 10·31-s + 33-s + 6·37-s − 6·39-s − 5·41-s + 4·43-s − 47-s + 49-s − 3·53-s + 7·57-s − 3·59-s + 9·61-s − 63-s − 4·67-s + 69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s + 1.60·19-s − 0.218·21-s + 0.208·23-s − 25-s + 0.192·27-s − 1.85·29-s − 1.79·31-s + 0.174·33-s + 0.986·37-s − 0.960·39-s − 0.780·41-s + 0.609·43-s − 0.145·47-s + 1/7·49-s − 0.412·53-s + 0.927·57-s − 0.390·59-s + 1.15·61-s − 0.125·63-s − 0.488·67-s + 0.120·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.86904722670008767540844365935, −7.47703773295219669660393661950, −6.91775809382163379906031392064, −5.73014744848908738634973527400, −5.19961660243052959946652038549, −4.12989577087164664443168788286, −3.40081066006281755679070730676, −2.53705933887033350003055850950, −1.59909781423144554471050104542, 0,
1.59909781423144554471050104542, 2.53705933887033350003055850950, 3.40081066006281755679070730676, 4.12989577087164664443168788286, 5.19961660243052959946652038549, 5.73014744848908738634973527400, 6.91775809382163379906031392064, 7.47703773295219669660393661950, 7.86904722670008767540844365935
