| L(s) = 1 | + (−12.2 + 10.2i)2-s + (−33.7 − 12.2i)3-s + (44.4 − 252. i)4-s + (−438. − 2.48e3i)5-s + (539. − 196. i)6-s + (2.23e3 − 3.86e3i)7-s + (2.04e3 + 3.54e3i)8-s + (−1.40e4 − 1.18e4i)9-s + (3.09e4 + 2.59e4i)10-s + (1.77e4 + 3.07e4i)11-s + (−4.59e3 + 7.95e3i)12-s + (−177. + 64.4i)13-s + (1.24e4 + 7.03e4i)14-s + (−1.57e4 + 8.92e4i)15-s + (−6.15e4 − 2.24e4i)16-s + (−4.96e4 + 4.16e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (−0.240 − 0.0874i)3-s + (0.0868 − 0.492i)4-s + (−0.313 − 1.77i)5-s + (0.169 − 0.0618i)6-s + (0.351 − 0.608i)7-s + (0.176 + 0.306i)8-s + (−0.715 − 0.600i)9-s + (0.978 + 0.821i)10-s + (0.365 + 0.633i)11-s + (−0.0639 + 0.110i)12-s + (−0.00171 + 0.000626i)13-s + (0.0863 + 0.489i)14-s + (−0.0802 + 0.454i)15-s + (−0.234 − 0.0855i)16-s + (−0.144 + 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.121i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0275832 + 0.453302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0275832 + 0.453302i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (12.2 - 10.2i)T \) |
| 19 | \( 1 + (-1.57e5 + 5.45e5i)T \) |
| good | 3 | \( 1 + (33.7 + 12.2i)T + (1.50e4 + 1.26e4i)T^{2} \) |
| 5 | \( 1 + (438. + 2.48e3i)T + (-1.83e6 + 6.68e5i)T^{2} \) |
| 7 | \( 1 + (-2.23e3 + 3.86e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-1.77e4 - 3.07e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (177. - 64.4i)T + (8.12e9 - 6.81e9i)T^{2} \) |
| 17 | \( 1 + (4.96e4 - 4.16e4i)T + (2.05e10 - 1.16e11i)T^{2} \) |
| 23 | \( 1 + (1.38e5 - 7.84e5i)T + (-1.69e12 - 6.16e11i)T^{2} \) |
| 29 | \( 1 + (3.53e6 + 2.96e6i)T + (2.51e12 + 1.42e13i)T^{2} \) |
| 31 | \( 1 + (7.09e5 - 1.22e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 1.54e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-1.26e7 - 4.61e6i)T + (2.50e14 + 2.10e14i)T^{2} \) |
| 43 | \( 1 + (-5.37e5 - 3.04e6i)T + (-4.72e14 + 1.71e14i)T^{2} \) |
| 47 | \( 1 + (-2.83e7 - 2.37e7i)T + (1.94e14 + 1.10e15i)T^{2} \) |
| 53 | \( 1 + (-6.30e6 + 3.57e7i)T + (-3.10e15 - 1.12e15i)T^{2} \) |
| 59 | \( 1 + (-3.80e7 + 3.19e7i)T + (1.50e15 - 8.53e15i)T^{2} \) |
| 61 | \( 1 + (2.55e7 - 1.45e8i)T + (-1.09e16 - 3.99e15i)T^{2} \) |
| 67 | \( 1 + (7.07e7 + 5.93e7i)T + (4.72e15 + 2.67e16i)T^{2} \) |
| 71 | \( 1 + (7.32e5 + 4.15e6i)T + (-4.30e16 + 1.56e16i)T^{2} \) |
| 73 | \( 1 + (2.85e8 + 1.04e8i)T + (4.50e16 + 3.78e16i)T^{2} \) |
| 79 | \( 1 + (1.14e8 + 4.16e7i)T + (9.18e16 + 7.70e16i)T^{2} \) |
| 83 | \( 1 + (4.02e8 - 6.97e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-7.76e7 + 2.82e7i)T + (2.68e17 - 2.25e17i)T^{2} \) |
| 97 | \( 1 + (-7.64e8 + 6.41e8i)T + (1.32e17 - 7.48e17i)T^{2} \) |
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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
−13.66807266981904902968246362365, −12.38033239498998769532021775821, −11.32683873738451035072343711703, −9.488719611528323520486679604044, −8.646591441082042168230033389750, −7.34991450451636261802804652375, −5.58272744594669518812587572602, −4.32013308821848873090651971523, −1.32747760582436111066305466630, −0.21615931581927781558124948905,
2.24655126203659340311588063022, 3.47149429636799386176803403465, 5.85935212394758806799701416973, 7.33660792307581747314046104180, 8.640233735693740459742601768564, 10.36673632158364012777331907414, 11.13015263047515825761230670469, 11.96403352848603895485838226223, 13.96061018800275947938068796599, 14.80094177470555503372918984417
