| L(s) = 1 | + 4·5-s − 3·9-s + 6·13-s + 2·17-s + 11·25-s − 10·29-s + 2·37-s − 8·41-s − 12·45-s − 7·49-s + 14·53-s − 12·61-s + 24·65-s + 6·73-s + 9·81-s + 8·85-s − 16·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 18·117-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 9-s + 1.66·13-s + 0.485·17-s + 11/5·25-s − 1.85·29-s + 0.328·37-s − 1.24·41-s − 1.78·45-s − 49-s + 1.92·53-s − 1.53·61-s + 2.97·65-s + 0.702·73-s + 81-s + 0.867·85-s − 1.69·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.01392606045436, −13.55301254679852, −13.25403946966818, −12.86291756689129, −12.09639654422094, −11.56808229431402, −10.97645475976604, −10.69768510633376, −10.12516150145657, −9.470657414892005, −9.242495356906391, −8.604111524211836, −8.292610972006151, −7.529648874156425, −6.790799152063967, −6.241812531747468, −5.962035323160820, −5.331997807621423, −5.205664286552451, −4.064866410379466, −3.524898917828506, −2.890455521992017, −2.294066063699461, −1.567790918465143, −1.183937303738402, 0,
1.183937303738402, 1.567790918465143, 2.294066063699461, 2.890455521992017, 3.524898917828506, 4.064866410379466, 5.205664286552451, 5.331997807621423, 5.962035323160820, 6.241812531747468, 6.790799152063967, 7.529648874156425, 8.292610972006151, 8.604111524211836, 9.242495356906391, 9.470657414892005, 10.12516150145657, 10.69768510633376, 10.97645475976604, 11.56808229431402, 12.09639654422094, 12.86291756689129, 13.25403946966818, 13.55301254679852, 14.01392606045436
