| L(s) = 1 | + (−33.3 + 12.1i)2-s + (12.1 + 4.43i)3-s + (573. − 481. i)4-s + (−434. − 2.46e3i)5-s − 460.·6-s + (−1.47e3 − 8.36e3i)7-s + (−4.20e3 + 7.28e3i)8-s + (−1.49e4 − 1.25e4i)9-s + (4.43e4 + 7.69e4i)10-s + (6.97e3 − 1.20e4i)11-s + (9.12e3 − 3.32e3i)12-s + (8.44e4 − 7.09e4i)13-s + (1.50e5 + 2.61e5i)14-s + (5.63e3 − 3.19e4i)15-s + (−1.47e4 + 8.34e4i)16-s + (−4.60e4 − 3.86e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.47 + 0.536i)2-s + (0.0868 + 0.0316i)3-s + (1.12 − 0.940i)4-s + (−0.310 − 1.76i)5-s − 0.145·6-s + (−0.232 − 1.31i)7-s + (−0.363 + 0.628i)8-s + (−0.759 − 0.637i)9-s + (1.40 + 2.43i)10-s + (0.143 − 0.248i)11-s + (0.127 − 0.0462i)12-s + (0.820 − 0.688i)13-s + (1.04 + 1.81i)14-s + (0.0287 − 0.162i)15-s + (−0.0561 + 0.318i)16-s + (−0.133 − 0.112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0626i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0185052 - 0.590137i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0185052 - 0.590137i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-1.01e7 + 5.15e6i)T \) |
| good | 2 | \( 1 + (33.3 - 12.1i)T + (392. - 329. i)T^{2} \) |
| 3 | \( 1 + (-12.1 - 4.43i)T + (1.50e4 + 1.26e4i)T^{2} \) |
| 5 | \( 1 + (434. + 2.46e3i)T + (-1.83e6 + 6.68e5i)T^{2} \) |
| 7 | \( 1 + (1.47e3 + 8.36e3i)T + (-3.79e7 + 1.38e7i)T^{2} \) |
| 11 | \( 1 + (-6.97e3 + 1.20e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-8.44e4 + 7.09e4i)T + (1.84e9 - 1.04e10i)T^{2} \) |
| 17 | \( 1 + (4.60e4 + 3.86e4i)T + (2.05e10 + 1.16e11i)T^{2} \) |
| 19 | \( 1 + (-1.68e5 - 6.12e4i)T + (2.47e11 + 2.07e11i)T^{2} \) |
| 23 | \( 1 + (8.08e5 + 1.39e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-5.26e5 + 9.12e5i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 - 2.66e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + (-1.76e7 + 1.47e7i)T + (5.68e13 - 3.22e14i)T^{2} \) |
| 43 | \( 1 + 2.57e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.38e7 - 2.40e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (1.04e7 - 5.93e7i)T + (-3.10e15 - 1.12e15i)T^{2} \) |
| 59 | \( 1 + (-2.43e7 + 1.38e8i)T + (-8.14e15 - 2.96e15i)T^{2} \) |
| 61 | \( 1 + (9.79e7 - 8.22e7i)T + (2.03e15 - 1.15e16i)T^{2} \) |
| 67 | \( 1 + (1.87e7 + 1.06e8i)T + (-2.55e16 + 9.30e15i)T^{2} \) |
| 71 | \( 1 + (2.33e8 + 8.50e7i)T + (3.51e16 + 2.94e16i)T^{2} \) |
| 73 | \( 1 - 4.04e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (-1.07e8 - 6.07e8i)T + (-1.12e17 + 4.09e16i)T^{2} \) |
| 83 | \( 1 + (1.24e8 + 1.04e8i)T + (3.24e16 + 1.84e17i)T^{2} \) |
| 89 | \( 1 + (5.62e7 - 3.19e8i)T + (-3.29e17 - 1.19e17i)T^{2} \) |
| 97 | \( 1 + (-5.44e8 - 9.43e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−13.79723902617938602193269968348, −12.52115950220291245981670555417, −10.92323616464531732233420153573, −9.561029448212653899938460495294, −8.603889013545884497683581570092, −7.79279606935234590655217338854, −6.08973403343952981822364259767, −4.08314090421409845715650394681, −0.933623420136080989630671258925, −0.47199285877962267838951694401,
2.08060003761677334843501183074, 3.04581527249878544716343278740, 6.15814282977181723170236963010, 7.60417265987908999889721515105, 8.783773178238187084366214760069, 9.993163761050273399774198647814, 11.23574415253674507074511654590, 11.69062716018182679437606375446, 13.93500617711422457861253435320, 15.13463846381413159085215436548
