| L(s) = 1 | + (15.0 + 5.47i)2-s + (26.9 + 152. i)3-s + (196. + 164. i)4-s + (−1.71e3 + 1.43e3i)5-s + (−431. + 2.44e3i)6-s + (−910. + 1.57e3i)7-s + (2.04e3 + 3.54e3i)8-s + (−4.14e3 + 1.50e3i)9-s + (−3.36e4 + 1.22e4i)10-s + (−4.01e4 − 6.94e4i)11-s + (−1.98e4 + 3.44e4i)12-s + (−6.69e3 + 3.79e4i)13-s + (−2.23e4 + 1.87e4i)14-s + (−2.65e5 − 2.22e5i)15-s + (1.13e4 + 6.45e4i)16-s + (6.26e5 + 2.28e5i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.192 + 1.08i)3-s + (0.383 + 0.321i)4-s + (−1.22 + 1.02i)5-s + (−0.135 + 0.770i)6-s + (−0.143 + 0.248i)7-s + (0.176 + 0.306i)8-s + (−0.210 + 0.0765i)9-s + (−1.06 + 0.386i)10-s + (−0.825 − 1.43i)11-s + (−0.276 + 0.479i)12-s + (−0.0649 + 0.368i)13-s + (−0.155 + 0.130i)14-s + (−1.35 − 1.13i)15-s + (0.0434 + 0.246i)16-s + (1.81 + 0.662i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.347401 - 1.33188i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.347401 - 1.33188i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-15.0 - 5.47i)T \) |
| 19 | \( 1 + (4.44e5 + 3.53e5i)T \) |
| good | 3 | \( 1 + (-26.9 - 152. i)T + (-1.84e4 + 6.73e3i)T^{2} \) |
| 5 | \( 1 + (1.71e3 - 1.43e3i)T + (3.39e5 - 1.92e6i)T^{2} \) |
| 7 | \( 1 + (910. - 1.57e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (4.01e4 + 6.94e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (6.69e3 - 3.79e4i)T + (-9.96e9 - 3.62e9i)T^{2} \) |
| 17 | \( 1 + (-6.26e5 - 2.28e5i)T + (9.08e10 + 7.62e10i)T^{2} \) |
| 23 | \( 1 + (2.89e5 + 2.42e5i)T + (3.12e11 + 1.77e12i)T^{2} \) |
| 29 | \( 1 + (6.88e6 - 2.50e6i)T + (1.11e13 - 9.32e12i)T^{2} \) |
| 31 | \( 1 + (3.25e6 - 5.64e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 1.53e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (1.73e5 + 9.84e5i)T + (-3.07e14 + 1.11e14i)T^{2} \) |
| 43 | \( 1 + (-9.41e6 + 7.90e6i)T + (8.72e13 - 4.94e14i)T^{2} \) |
| 47 | \( 1 + (1.97e7 - 7.17e6i)T + (8.57e14 - 7.19e14i)T^{2} \) |
| 53 | \( 1 + (-1.09e7 - 9.15e6i)T + (5.72e14 + 3.24e15i)T^{2} \) |
| 59 | \( 1 + (2.17e6 + 7.90e5i)T + (6.63e15 + 5.56e15i)T^{2} \) |
| 61 | \( 1 + (-1.55e8 - 1.30e8i)T + (2.03e15 + 1.15e16i)T^{2} \) |
| 67 | \( 1 + (-1.39e8 + 5.08e7i)T + (2.08e16 - 1.74e16i)T^{2} \) |
| 71 | \( 1 + (1.56e8 - 1.31e8i)T + (7.96e15 - 4.51e16i)T^{2} \) |
| 73 | \( 1 + (2.81e6 + 1.59e7i)T + (-5.53e16 + 2.01e16i)T^{2} \) |
| 79 | \( 1 + (4.82e6 + 2.73e7i)T + (-1.12e17 + 4.09e16i)T^{2} \) |
| 83 | \( 1 + (1.53e8 - 2.66e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (2.83e7 - 1.60e8i)T + (-3.29e17 - 1.19e17i)T^{2} \) |
| 97 | \( 1 + (2.27e8 + 8.28e7i)T + (5.82e17 + 4.88e17i)T^{2} \) |
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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
−15.01008632361733339306453520583, −14.27810842854724778665831648515, −12.57243024426013268209268238142, −11.17752965431649360692654166209, −10.45616893479996684208329094697, −8.555656470058579632772998100210, −7.22660093732919401755102895476, −5.51663700147716043190278801910, −3.80852013720398199576024164829, −3.16606698627260109938964479604,
0.38328766950597285017280172970, 1.87611495084902396115768442929, 3.85851840454225235741654764943, 5.27354255981432793768213764911, 7.37676599713565798729362905288, 7.896655247772050972551055898589, 9.964657282107667372082866616147, 11.78967663154465949360328190922, 12.60174722680864796059564527657, 13.08876719170220977678322270128
