| L(s) = 1 | + 0.879·3-s − 0.694·5-s + 7-s − 2.22·9-s − 4.36·11-s − 3.71·13-s − 0.610·15-s + 2.53·17-s + 2.29·19-s + 0.879·21-s − 3.94·23-s − 4.51·25-s − 4.59·27-s − 0.241·29-s − 5.38·31-s − 3.84·33-s − 0.694·35-s − 11.1·37-s − 3.26·39-s − 41-s + 4.98·43-s + 1.54·45-s + 9.92·47-s + 49-s + 2.22·51-s + 12.8·53-s + 3.03·55-s + ⋯ |
| L(s) = 1 | + 0.507·3-s − 0.310·5-s + 0.377·7-s − 0.742·9-s − 1.31·11-s − 1.03·13-s − 0.157·15-s + 0.614·17-s + 0.525·19-s + 0.191·21-s − 0.822·23-s − 0.903·25-s − 0.884·27-s − 0.0447·29-s − 0.967·31-s − 0.668·33-s − 0.117·35-s − 1.82·37-s − 0.523·39-s − 0.156·41-s + 0.760·43-s + 0.230·45-s + 1.44·47-s + 0.142·49-s + 0.311·51-s + 1.76·53-s + 0.409·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 0.879T + 3T^{2} \) |
| 5 | \( 1 + 0.694T + 5T^{2} \) |
| 11 | \( 1 + 4.36T + 11T^{2} \) |
| 13 | \( 1 + 3.71T + 13T^{2} \) |
| 17 | \( 1 - 2.53T + 17T^{2} \) |
| 19 | \( 1 - 2.29T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 + 0.241T + 29T^{2} \) |
| 31 | \( 1 + 5.38T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 43 | \( 1 - 4.98T + 43T^{2} \) |
| 47 | \( 1 - 9.92T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 3.38T + 59T^{2} \) |
| 61 | \( 1 + 6.24T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 + 2.61T + 73T^{2} \) |
| 79 | \( 1 - 1.91T + 79T^{2} \) |
| 83 | \( 1 - 1.51T + 83T^{2} \) |
| 89 | \( 1 + 2.32T + 89T^{2} \) |
| 97 | \( 1 - 4.26T + 97T^{2} \) |
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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
−9.332584409533458254803556527133, −8.480826330939252212195663794203, −7.69247962891737510788504612030, −7.32734396084236862853926910226, −5.70338405891256515122302297527, −5.26943880588327811079493016195, −4.00607624713069490703381911459, −2.96235071489821099263400895465, −2.07586807629839194786398762236, 0,
2.07586807629839194786398762236, 2.96235071489821099263400895465, 4.00607624713069490703381911459, 5.26943880588327811079493016195, 5.70338405891256515122302297527, 7.32734396084236862853926910226, 7.69247962891737510788504612030, 8.480826330939252212195663794203, 9.332584409533458254803556527133
