| L(s) = 1 | + (−7.72 + 13.3i)2-s + (11.6 − 45.2i)3-s + (−55.3 − 95.8i)4-s + (15.6 + 27.0i)5-s + (516. + 505. i)6-s + (−171.5 + 297. i)7-s − 267.·8-s + (−1.91e3 − 1.05e3i)9-s − 483.·10-s + (337. − 585. i)11-s + (−4.98e3 + 1.39e3i)12-s + (1.48e3 + 2.57e3i)13-s + (−2.64e3 − 4.58e3i)14-s + (1.40e3 − 393. i)15-s + (9.15e3 − 1.58e4i)16-s + 1.32e4·17-s + ⋯ |
| L(s) = 1 | + (−0.682 + 1.18i)2-s + (0.248 − 0.968i)3-s + (−0.432 − 0.748i)4-s + (0.0559 + 0.0969i)5-s + (0.975 + 0.955i)6-s + (−0.188 + 0.327i)7-s − 0.185·8-s + (−0.876 − 0.481i)9-s − 0.152·10-s + (0.0765 − 0.132i)11-s + (−0.832 + 0.232i)12-s + (0.187 + 0.325i)13-s + (−0.258 − 0.446i)14-s + (0.107 − 0.0301i)15-s + (0.558 − 0.967i)16-s + 0.653·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.18831 + 0.569470i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18831 + 0.569470i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-11.6 + 45.2i)T \) |
| 7 | \( 1 + (171.5 - 297. i)T \) |
| good | 2 | \( 1 + (7.72 - 13.3i)T + (-64 - 110. i)T^{2} \) |
| 5 | \( 1 + (-15.6 - 27.0i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-337. + 585. i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-1.48e3 - 2.57e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 - 1.32e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.03e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-1.40e3 - 2.42e3i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (5.03e4 - 8.71e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-7.03e4 - 1.21e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 4.52e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-3.45e5 - 5.98e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.67e5 + 2.90e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-1.54e5 + 2.67e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 - 1.56e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-8.73e5 - 1.51e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.39e6 + 2.41e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-4.55e5 - 7.89e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 3.70e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.89e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-8.17e4 + 1.41e5i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.24e6 + 2.16e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + 9.04e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-3.35e5 + 5.80e5i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.99302169608105618034274078907, −12.61221994530803650541787527282, −11.58565274292716038556293076528, −9.657669445182808520031022145316, −8.619971562780052897657588948240, −7.63641225632915174243697567128, −6.65179832303583712890898730669, −5.61520158044253037487797790232, −2.95293719831774267110986679604, −0.919301310875239249746706992953,
0.853619748898797986459812733780, 2.70593074478762877329669141034, 3.86698317490954763733624079648, 5.62976010694115014825037030206, 7.85945538588442760826153938283, 9.256841778243820049829967934503, 9.853553059919718244912351930442, 10.89115682947971747425101685146, 11.76270541847321429224440234060, 13.13398468116418885341654124146
