| L(s) = 1 | + (−1.84 − 1.06i)2-s + (−6.34 + 14.2i)3-s + (−13.7 − 23.7i)4-s + (47.5 − 82.3i)5-s + (26.8 − 19.5i)6-s + (−95.0 − 88.1i)7-s + 126. i·8-s + (−162. − 180. i)9-s + (−175. + 101. i)10-s + (113. − 65.3i)11-s + (425. − 44.7i)12-s + 14.6i·13-s + (81.5 + 264. i)14-s + (870. + 1.19e3i)15-s + (−304. + 526. i)16-s + (−320. − 554. i)17-s + ⋯ |
| L(s) = 1 | + (−0.326 − 0.188i)2-s + (−0.406 + 0.913i)3-s + (−0.428 − 0.742i)4-s + (0.850 − 1.47i)5-s + (0.305 − 0.221i)6-s + (−0.733 − 0.680i)7-s + 0.700i·8-s + (−0.669 − 0.743i)9-s + (−0.555 + 0.320i)10-s + (0.282 − 0.162i)11-s + (0.853 − 0.0896i)12-s + 0.0240i·13-s + (0.111 + 0.360i)14-s + (0.999 + 1.37i)15-s + (−0.296 + 0.514i)16-s + (−0.268 − 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.510934 - 0.655268i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.510934 - 0.655268i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (6.34 - 14.2i)T \) |
| 7 | \( 1 + (95.0 + 88.1i)T \) |
| good | 2 | \( 1 + (1.84 + 1.06i)T + (16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-47.5 + 82.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-113. + 65.3i)T + (8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 14.6iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (320. + 554. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (469. + 271. i)T + (1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-3.54e3 - 2.04e3i)T + (3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 3.59e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (2.51e3 - 1.45e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-2.17e3 + 3.76e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.09e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (4.40e3 - 7.63e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-6.04e3 + 3.48e3i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (2.00e4 + 3.46e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.87e4 + 1.66e4i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (3.21e3 + 5.57e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 3.24e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (1.10e4 - 6.36e3i)T + (1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (4.29e3 - 7.44e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 3.16e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-1.59e4 + 2.76e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.43e5iT - 8.58e9T^{2} \) |
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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
−16.92237080167121133636097939453, −15.77549776927006782138875780646, −14.07924942569471484601033389217, −12.90230879196452087023161548043, −10.96843045107919747805474399232, −9.580586036950743521178487384354, −9.103037450713567902641611326012, −5.86815804464099540035298131287, −4.59047000726935411462090729969, −0.73224023470626463957418727283,
2.75093634681424344529541063981, 6.18518434717508238661733653061, 7.20117516869998343412908003217, 9.070359794291844210630761673491, 10.74102591816692111039834904013, 12.42248327903245527892680721345, 13.41401002238476661992270308818, 14.78653822830042142154226185726, 16.65752948073496991284745560223, 17.72506852641497002187677778205
