| L(s) = 1 | + 4·7-s + 4·11-s + 4·13-s + 2·17-s + 4·19-s − 8·23-s − 5·25-s + 8·29-s + 4·31-s − 4·37-s − 6·41-s − 4·43-s − 8·47-s + 9·49-s + 8·53-s − 12·59-s + 12·61-s − 12·67-s + 8·71-s − 6·73-s + 16·77-s + 4·79-s − 4·83-s + 6·89-s + 16·91-s − 2·97-s + 8·101-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.20·11-s + 1.10·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s − 25-s + 1.48·29-s + 0.718·31-s − 0.657·37-s − 0.937·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 1.09·53-s − 1.56·59-s + 1.53·61-s − 1.46·67-s + 0.949·71-s − 0.702·73-s + 1.82·77-s + 0.450·79-s − 0.439·83-s + 0.635·89-s + 1.67·91-s − 0.203·97-s + 0.796·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.520523471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.520523471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 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
−8.770780213851605268751258605897, −8.273401806148982335189253098083, −7.66603558879594006866408095630, −6.59361365188310900976865652323, −5.90734355349400077940850184592, −4.98531549119289558181541630311, −4.16197372918625835514829553019, −3.38591647618518318996390357483, −1.87178167433974199275141976633, −1.18014352853545560604310847207,
1.18014352853545560604310847207, 1.87178167433974199275141976633, 3.38591647618518318996390357483, 4.16197372918625835514829553019, 4.98531549119289558181541630311, 5.90734355349400077940850184592, 6.59361365188310900976865652323, 7.66603558879594006866408095630, 8.273401806148982335189253098083, 8.770780213851605268751258605897
