| L(s) = 1 | + (8 + 13.8i)2-s + (−127. + 221. i)4-s + (419. + 727. i)5-s + (−6.25e3 − 1.10e3i)7-s − 4.09e3·8-s + (−6.71e3 + 1.16e4i)10-s + (−2.06e4 + 3.57e4i)11-s − 9.19e4·13-s + (−3.46e4 − 9.55e4i)14-s + (−3.27e4 − 5.67e4i)16-s + (1.54e5 − 2.67e5i)17-s + (3.96e5 + 6.86e5i)19-s − 2.14e5·20-s − 6.61e5·22-s + (−4.66e5 − 8.08e5i)23-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.300 + 0.520i)5-s + (−0.984 − 0.174i)7-s − 0.353·8-s + (−0.212 + 0.367i)10-s + (−0.425 + 0.737i)11-s − 0.893·13-s + (−0.241 − 0.664i)14-s + (−0.125 − 0.216i)16-s + (0.448 − 0.776i)17-s + (0.697 + 1.20i)19-s − 0.300·20-s − 0.601·22-s + (−0.347 − 0.602i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9252814048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9252814048\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8 - 13.8i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (6.25e3 + 1.10e3i)T \) |
| good | 5 | \( 1 + (-419. - 727. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (2.06e4 - 3.57e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 9.19e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-1.54e5 + 2.67e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-3.96e5 - 6.86e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (4.66e5 + 8.08e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 5.92e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-3.31e6 + 5.74e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-9.40e6 - 1.62e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 3.06e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.69e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (2.04e7 + 3.54e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-3.67e7 + 6.36e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-1.10e7 + 1.91e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (3.55e7 + 6.16e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.09e8 + 1.90e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 1.77e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + (2.18e7 - 3.78e7i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-3.03e7 - 5.25e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.88e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (4.38e7 + 7.59e7i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.15e9T + 7.60e17T^{2} \) |
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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
−11.75116080951957309457971226954, −10.02066011699754717406442120812, −9.721278883784236635799895874959, −7.958594299811961956131281008146, −7.06697776948715242244407195043, −6.10440341094739706228624337166, −4.91151011121380049234994719801, −3.49264826273014752108281079337, −2.34545298419041419624162197990, −0.22291117692148922008855220622,
1.02566675684459845945643045145, 2.54400324607107325105768465168, 3.57594294096904773912875918703, 5.09301770626057943588236652382, 5.94864440638205597059713190835, 7.41620119997410083338194544258, 8.969063986203177561610094868837, 9.656482556287503781646396597914, 10.74660886669679464000883178336, 11.86565233905450694466365993995
