L(s) = 1 | − 2·4-s − 5-s + 4·7-s − 3·11-s + 13-s + 4·16-s − 19-s + 2·20-s + 6·23-s + 25-s − 8·28-s + 3·29-s − 8·31-s − 4·35-s − 2·37-s + 9·41-s + 8·43-s + 6·44-s − 12·47-s + 9·49-s − 2·52-s + 12·53-s + 3·55-s − 3·59-s − 5·61-s − 8·64-s − 65-s + ⋯ |
L(s) = 1 | − 4-s − 0.447·5-s + 1.51·7-s − 0.904·11-s + 0.277·13-s + 16-s − 0.229·19-s + 0.447·20-s + 1.25·23-s + 1/5·25-s − 1.51·28-s + 0.557·29-s − 1.43·31-s − 0.676·35-s − 0.328·37-s + 1.40·41-s + 1.21·43-s + 0.904·44-s − 1.75·47-s + 9/7·49-s − 0.277·52-s + 1.64·53-s + 0.404·55-s − 0.390·59-s − 0.640·61-s − 64-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.34874812182393, −13.00494755033819, −12.64927838567888, −12.07982427460274, −11.48051369258114, −11.02622146712951, −10.70726075492545, −10.29850845260420, −9.443442583748133, −9.166736551775385, −8.540910427991789, −8.202470987780876, −7.843783763975294, −7.287597999511911, −6.840437516053022, −5.822621592958369, −5.482085710096963, −5.026966033704053, −4.515369481454336, −4.159145681874378, −3.483434639638744, −2.840958604388856, −2.159203380722034, −1.365188295506977, −0.8401792269006778, 0,
0.8401792269006778, 1.365188295506977, 2.159203380722034, 2.840958604388856, 3.483434639638744, 4.159145681874378, 4.515369481454336, 5.026966033704053, 5.482085710096963, 5.822621592958369, 6.840437516053022, 7.287597999511911, 7.843783763975294, 8.202470987780876, 8.540910427991789, 9.166736551775385, 9.443442583748133, 10.29850845260420, 10.70726075492545, 11.02622146712951, 11.48051369258114, 12.07982427460274, 12.64927838567888, 13.00494755033819, 13.34874812182393