| L(s) = 1 | + 5-s − 4·11-s + 13-s + 8·17-s + 2·19-s − 4·23-s + 25-s + 8·29-s − 10·31-s + 6·37-s + 6·41-s − 8·43-s − 8·47-s + 12·53-s − 4·55-s − 4·59-s − 10·61-s + 65-s + 2·67-s − 6·73-s + 12·79-s − 4·83-s + 8·85-s − 6·89-s + 2·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s + 0.277·13-s + 1.94·17-s + 0.458·19-s − 0.834·23-s + 1/5·25-s + 1.48·29-s − 1.79·31-s + 0.986·37-s + 0.937·41-s − 1.21·43-s − 1.16·47-s + 1.64·53-s − 0.539·55-s − 0.520·59-s − 1.28·61-s + 0.124·65-s + 0.244·67-s − 0.702·73-s + 1.35·79-s − 0.439·83-s + 0.867·85-s − 0.635·89-s + 0.205·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.01125698783951, −12.87031539991372, −12.14880834796404, −11.93722294965686, −11.29150549231400, −10.72382668417831, −10.29369901893664, −10.01437151041222, −9.496490438585906, −9.052668961642388, −8.213599181535524, −8.091608381648814, −7.523887801009526, −7.117118192093033, −6.348138211918547, −5.895955188018661, −5.472063992566014, −5.089677840351044, −4.480535269759267, −3.753217043319097, −3.215470390020977, −2.796700155328572, −2.127368309380737, −1.449811198975444, −0.8708989368687804, 0,
0.8708989368687804, 1.449811198975444, 2.127368309380737, 2.796700155328572, 3.215470390020977, 3.753217043319097, 4.480535269759267, 5.089677840351044, 5.472063992566014, 5.895955188018661, 6.348138211918547, 7.117118192093033, 7.523887801009526, 8.091608381648814, 8.213599181535524, 9.052668961642388, 9.496490438585906, 10.01437151041222, 10.29369901893664, 10.72382668417831, 11.29150549231400, 11.93722294965686, 12.14880834796404, 12.87031539991372, 13.01125698783951
