| L(s) = 1 | + (−4.5 + 2.59i)3-s + (−31.0 − 17.9i)5-s + (−41.3 − 26.2i)7-s + (13.5 − 23.3i)9-s + (21.0 + 36.5i)11-s + 22.7i·13-s + 186.·15-s + (111. − 64.3i)17-s + (338. + 195. i)19-s + (254. + 10.6i)21-s + (−331. + 574. i)23-s + (329. + 570. i)25-s + 140. i·27-s − 604.·29-s + (1.10e3 − 635. i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 + 0.288i)3-s + (−1.24 − 0.716i)5-s + (−0.844 − 0.535i)7-s + (0.166 − 0.288i)9-s + (0.174 + 0.301i)11-s + 0.134i·13-s + 0.827·15-s + (0.385 − 0.222i)17-s + (0.938 + 0.541i)19-s + (0.576 + 0.0241i)21-s + (−0.626 + 1.08i)23-s + (0.527 + 0.913i)25-s + 0.192i·27-s − 0.718·29-s + (1.14 − 0.661i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7619555565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7619555565\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.5 - 2.59i)T \) |
| 7 | \( 1 + (41.3 + 26.2i)T \) |
| good | 5 | \( 1 + (31.0 + 17.9i)T + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (-21.0 - 36.5i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 22.7iT - 2.85e4T^{2} \) |
| 17 | \( 1 + (-111. + 64.3i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-338. - 195. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (331. - 574. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + 604.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (-1.10e3 + 635. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (1.06e3 - 1.85e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + 244. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.48e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (-114. - 66.1i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-481. - 833. i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-5.54e3 + 3.20e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (524. + 302. i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-1.59e3 - 2.76e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 6.48e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-526. + 304. i)T + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-596. + 1.03e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 4.57e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (9.45e3 + 5.46e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + 9.24e3iT - 8.85e7T^{2} \) |
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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
−12.03748970269355850906477820577, −11.54804352869362569714564865478, −10.16712240625025203374096517740, −9.371602327400640407772869861928, −7.989558375191377206257228842778, −7.11980102711111245195144524087, −5.69710025173380811257671494946, −4.36497760417418593340106844988, −3.50030044325120931865963635613, −0.892385152196412788346291630491,
0.43788530602779012103677409357, 2.79914241190998931366607112721, 3.97560325182182917392623017877, 5.64005010935834659579413090283, 6.74783297794313144856827567241, 7.61347150970958472208918699086, 8.810376697639588450883867727072, 10.16279711966849421898029647063, 11.13254298943547762542237069541, 11.99114395633572204543842576123
