| L(s) = 1 | + 3-s + 5-s − 3·7-s − 2·9-s − 2·11-s − 13-s + 15-s + 4·17-s − 6·19-s − 3·21-s − 6·23-s − 4·25-s − 5·27-s + 8·29-s + 4·31-s − 2·33-s − 3·35-s − 8·37-s − 39-s − 10·41-s − 4·43-s − 2·45-s + 9·47-s + 2·49-s + 4·51-s − 2·53-s − 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s − 2/3·9-s − 0.603·11-s − 0.277·13-s + 0.258·15-s + 0.970·17-s − 1.37·19-s − 0.654·21-s − 1.25·23-s − 4/5·25-s − 0.962·27-s + 1.48·29-s + 0.718·31-s − 0.348·33-s − 0.507·35-s − 1.31·37-s − 0.160·39-s − 1.56·41-s − 0.609·43-s − 0.298·45-s + 1.31·47-s + 2/7·49-s + 0.560·51-s − 0.274·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−9.310336306909389057164283545641, −8.423348769693386380835018190291, −7.88526745996423744926027143743, −6.63518697987213341805199647856, −6.05892058215090753538341774685, −5.11838942883562842777257790218, −3.80042590498607980001670074245, −2.95547541728557219784982307746, −2.07845972721325403340873886517, 0,
2.07845972721325403340873886517, 2.95547541728557219784982307746, 3.80042590498607980001670074245, 5.11838942883562842777257790218, 6.05892058215090753538341774685, 6.63518697987213341805199647856, 7.88526745996423744926027143743, 8.423348769693386380835018190291, 9.310336306909389057164283545641
