| L(s) = 1 | + (−14.2 + 6.21i)3-s + (82.0 + 68.8i)5-s + (4.01 + 1.46i)7-s + (165. − 177. i)9-s + (339. − 285. i)11-s + (−111. + 633. i)13-s + (−1.60e3 − 474. i)15-s + (243. − 421. i)17-s + (270. + 468. i)19-s + (−66.4 + 4.04i)21-s + (493. − 179. i)23-s + (1.45e3 + 8.22e3i)25-s + (−1.26e3 + 3.56e3i)27-s + (1.29e3 + 7.34e3i)29-s + (−2.45e3 + 891. i)31-s + ⋯ |
| L(s) = 1 | + (−0.917 + 0.398i)3-s + (1.46 + 1.23i)5-s + (0.0309 + 0.0112i)7-s + (0.682 − 0.731i)9-s + (0.846 − 0.710i)11-s + (−0.183 + 1.03i)13-s + (−1.83 − 0.544i)15-s + (0.204 − 0.354i)17-s + (0.171 + 0.297i)19-s + (−0.0328 + 0.00200i)21-s + (0.194 − 0.0708i)23-s + (0.464 + 2.63i)25-s + (−0.334 + 0.942i)27-s + (0.285 + 1.62i)29-s + (−0.457 + 0.166i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0662 - 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0662 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.20681 + 1.28958i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20681 + 1.28958i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (14.2 - 6.21i)T \) |
| good | 5 | \( 1 + (-82.0 - 68.8i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (-4.01 - 1.46i)T + (1.28e4 + 1.08e4i)T^{2} \) |
| 11 | \( 1 + (-339. + 285. i)T + (2.79e4 - 1.58e5i)T^{2} \) |
| 13 | \( 1 + (111. - 633. i)T + (-3.48e5 - 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-243. + 421. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-270. - 468. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-493. + 179. i)T + (4.93e6 - 4.13e6i)T^{2} \) |
| 29 | \( 1 + (-1.29e3 - 7.34e3i)T + (-1.92e7 + 7.01e6i)T^{2} \) |
| 31 | \( 1 + (2.45e3 - 891. i)T + (2.19e7 - 1.84e7i)T^{2} \) |
| 37 | \( 1 + (-3.50e3 + 6.06e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (2.74e3 - 1.55e4i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + (-1.21e3 + 1.01e3i)T + (2.55e7 - 1.44e8i)T^{2} \) |
| 47 | \( 1 + (2.73e4 + 9.94e3i)T + (1.75e8 + 1.47e8i)T^{2} \) |
| 53 | \( 1 + 1.46e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-3.07e4 - 2.58e4i)T + (1.24e8 + 7.04e8i)T^{2} \) |
| 61 | \( 1 + (-1.38e4 - 5.04e3i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-2.94e3 + 1.66e4i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-2.96e4 + 5.14e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (9.24e3 + 1.60e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-1.61e4 - 9.17e4i)T + (-2.89e9 + 1.05e9i)T^{2} \) |
| 83 | \( 1 + (7.71e3 + 4.37e4i)T + (-3.70e9 + 1.34e9i)T^{2} \) |
| 89 | \( 1 + (-9.61e3 - 1.66e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-5.45e4 + 4.58e4i)T + (1.49e9 - 8.45e9i)T^{2} \) |
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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
−13.09604561033796685601772442552, −11.65225113098481819031213135433, −10.93453104646577898621220345859, −9.923746420887822836459370524482, −9.176347654760030089902273770748, −6.88979996585580299390471275523, −6.32876686909297765546580644318, −5.17866528894663600708238604064, −3.36801077638161024580890337527, −1.57076875554616733808164367966,
0.812543584729688332808939343278, 1.94411304531773382761637594747, 4.64569695982230488719691225734, 5.58232102037576933409514014710, 6.51554933546629763990153961440, 8.087292398602212504068999959000, 9.511219253286230928643007254981, 10.16348156137179511894566228421, 11.63635290072483298538435682613, 12.71974703614865950439673030592
