| L(s) = 1 | + (−0.724 − 1.21i)2-s + (−1.03 + 1.38i)3-s + (−0.948 + 1.76i)4-s + (1.94 − 1.10i)5-s + (2.43 + 0.251i)6-s − 2.63·7-s + (2.82 − 0.124i)8-s + (−0.853 − 2.87i)9-s + (−2.74 − 1.56i)10-s + (0.908 + 0.660i)11-s + (−1.46 − 3.14i)12-s + (4.15 + 5.72i)13-s + (1.90 + 3.19i)14-s + (−0.487 + 3.84i)15-s + (−2.19 − 3.34i)16-s + (−1.53 + 4.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.512 − 0.858i)2-s + (−0.598 + 0.801i)3-s + (−0.474 + 0.880i)4-s + (0.870 − 0.492i)5-s + (0.994 + 0.102i)6-s − 0.994·7-s + (0.999 − 0.0439i)8-s + (−0.284 − 0.958i)9-s + (−0.868 − 0.494i)10-s + (0.273 + 0.199i)11-s + (−0.421 − 0.906i)12-s + (1.15 + 1.58i)13-s + (0.509 + 0.853i)14-s + (−0.125 + 0.992i)15-s + (−0.549 − 0.835i)16-s + (−0.372 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.815116 + 0.163939i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.815116 + 0.163939i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.724 + 1.21i)T \) |
| 3 | \( 1 + (1.03 - 1.38i)T \) |
| 5 | \( 1 + (-1.94 + 1.10i)T \) |
| good | 7 | \( 1 + 2.63T + 7T^{2} \) |
| 11 | \( 1 + (-0.908 - 0.660i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-4.15 - 5.72i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.53 - 4.72i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-6.71 - 2.18i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.81 - 2.49i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.240 - 0.0782i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.71 - 1.53i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.57 + 3.53i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.13 - 2.93i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 7.40T + 43T^{2} \) |
| 47 | \( 1 + (-9.30 + 3.02i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.355 - 1.09i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (1.37 - 0.998i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (3.31 + 2.41i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.555 + 1.70i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.17 + 6.69i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.35 - 1.86i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-5.36 + 1.74i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (8.73 + 2.83i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (7.11 - 9.79i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-1.59 + 0.517i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.73115213952949346097113731478, −10.72825664467095486179612206484, −9.829868826291208361026879027887, −9.355646832173603454022105498457, −8.574592492934734301672435582144, −6.72920266524168658700542721045, −5.80563575851818275156453472506, −4.36618290656558358602145518453, −3.44390419345701235561847980683, −1.51500656089324346712121850909,
0.889006010944531200557675812500, 2.92278392885008721514905202087, 5.24478774752278569804219305223, 6.00045219969657394219759156669, 6.69831927939077055908766784899, 7.57526960535492091467922987384, 8.758177092079974151340326330741, 9.822195968192913160491286427276, 10.53145278152869345414611021734, 11.53111849974196240494452686864
