| L(s) = 1 | + (8.70 − 7.23i)2-s + (−45.0 − 77.9i)3-s + (23.4 − 125. i)4-s + (89.8 + 51.8i)5-s + (−955. − 353. i)6-s + (733. − 534. i)7-s + (−705. − 1.26e3i)8-s + (−2.96e3 + 5.12e3i)9-s + (1.15e3 − 198. i)10-s + (−1.38e3 + 799. i)11-s + (−1.08e4 + 3.83e3i)12-s + 5.06e3i·13-s + (2.51e3 − 9.95e3i)14-s − 9.33e3i·15-s + (−1.52e4 − 5.89e3i)16-s + (2.68e4 − 1.55e4i)17-s + ⋯ |
| L(s) = 1 | + (0.769 − 0.639i)2-s + (−0.962 − 1.66i)3-s + (0.183 − 0.983i)4-s + (0.321 + 0.185i)5-s + (−1.80 − 0.667i)6-s + (0.808 − 0.588i)7-s + (−0.487 − 0.873i)8-s + (−1.35 + 2.34i)9-s + (0.365 − 0.0626i)10-s + (−0.313 + 0.181i)11-s + (−1.81 + 0.641i)12-s + 0.639i·13-s + (0.245 − 0.969i)14-s − 0.714i·15-s + (−0.932 − 0.360i)16-s + (1.32 − 0.766i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.184742 + 1.79689i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.184742 + 1.79689i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8.70 + 7.23i)T \) |
| 7 | \( 1 + (-733. + 534. i)T \) |
| good | 3 | \( 1 + (45.0 + 77.9i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-89.8 - 51.8i)T + (3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (1.38e3 - 799. i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 - 5.06e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + (-2.68e4 + 1.55e4i)T + (2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-6.01e3 + 1.04e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (5.14e4 + 2.96e4i)T + (1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 - 8.74e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (2.73e4 + 4.73e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (9.31e4 - 1.61e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 4.24e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 3.08e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + (2.40e5 - 4.16e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-3.49e5 - 6.06e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-5.25e4 - 9.10e4i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.68e6 - 9.73e5i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-9.30e5 + 5.36e5i)T + (3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 4.90e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + (-4.45e6 + 2.57e6i)T + (5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (6.15e5 + 3.55e5i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 - 5.63e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-3.00e6 - 1.73e6i)T + (2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 4.80e6iT - 8.07e13T^{2} \) |
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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
−14.19118297982691859645290466156, −13.68960696442084258502963777563, −12.31113119005332626077622168283, −11.59337090041967938448733865622, −10.39081724487761629001802411202, −7.65130837332710868328730972186, −6.37020903240586759566034661104, −5.03282245128900180066758766612, −2.14734071096471765606974414286, −0.816843554467072637628674161670,
3.63537959858219247030160830792, 5.21571591407056618436472087632, 5.77131205410908237836387956368, 8.320720320650291644952267679985, 9.962066114780421556220057620162, 11.33500839438584297745226465172, 12.36209858309672092653284441276, 14.34375594443092855237824815619, 15.23958337573116170586879214427, 16.11220222906937795240390749584
