| L(s) = 1 | + 3·5-s − 7-s − 3·9-s − 2·11-s + 2·17-s + 3·19-s + 3·23-s + 4·25-s + 29-s + 9·31-s − 3·35-s + 6·37-s + 6·41-s + 9·43-s − 9·45-s − 47-s + 49-s + 5·53-s − 6·55-s + 4·59-s + 6·61-s + 3·63-s + 4·67-s − 10·71-s + 3·73-s + 2·77-s + 3·79-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 9-s − 0.603·11-s + 0.485·17-s + 0.688·19-s + 0.625·23-s + 4/5·25-s + 0.185·29-s + 1.61·31-s − 0.507·35-s + 0.986·37-s + 0.937·41-s + 1.37·43-s − 1.34·45-s − 0.145·47-s + 1/7·49-s + 0.686·53-s − 0.809·55-s + 0.520·59-s + 0.768·61-s + 0.377·63-s + 0.488·67-s − 1.18·71-s + 0.351·73-s + 0.227·77-s + 0.337·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.369241216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.369241216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−14.14505755573881, −13.52701174051859, −13.11462085293276, −12.78302031247953, −11.91919714937773, −11.71824778861670, −10.89324308849836, −10.55117114303856, −9.931723749856818, −9.562249609037577, −9.125673233078104, −8.510680336703814, −7.972445709145924, −7.401233640986795, −6.723045000180985, −6.069766181617084, −5.832609954264268, −5.283994348664517, −4.761380279222137, −3.931352826149192, −3.096348578605884, −2.610711027187988, −2.309319966528243, −1.185480226281621, −0.6593435501555223,
0.6593435501555223, 1.185480226281621, 2.309319966528243, 2.610711027187988, 3.096348578605884, 3.931352826149192, 4.761380279222137, 5.283994348664517, 5.832609954264268, 6.069766181617084, 6.723045000180985, 7.401233640986795, 7.972445709145924, 8.510680336703814, 9.125673233078104, 9.562249609037577, 9.931723749856818, 10.55117114303856, 10.89324308849836, 11.71824778861670, 11.91919714937773, 12.78302031247953, 13.11462085293276, 13.52701174051859, 14.14505755573881
