| L(s) = 1 | + (2.47 − 7.60i)2-s + (−23.1 − 71.2i)3-s + (−51.7 − 37.6i)4-s + 456.·5-s − 599.·6-s + (−1.20e3 − 873. i)7-s + (−414. + 300. i)8-s + (−2.77e3 + 2.01e3i)9-s + (1.12e3 − 3.47e3i)10-s + (−449. − 326. i)11-s + (−1.48e3 + 4.56e3i)12-s + (−2.53e3 − 7.80e3i)13-s + (−9.61e3 + 6.98e3i)14-s + (−1.05e4 − 3.25e4i)15-s + (1.26e3 + 3.89e3i)16-s + (2.47e4 − 1.79e4i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.495 − 1.52i)3-s + (−0.404 − 0.293i)4-s + 1.63·5-s − 1.13·6-s + (−1.32 − 0.962i)7-s + (−0.286 + 0.207i)8-s + (−1.26 + 0.921i)9-s + (0.356 − 1.09i)10-s + (−0.101 − 0.0740i)11-s + (−0.247 + 0.761i)12-s + (−0.320 − 0.985i)13-s + (−0.936 + 0.680i)14-s + (−0.808 − 2.48i)15-s + (0.0772 + 0.237i)16-s + (1.22 − 0.886i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 - 0.878i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.476 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.679058 + 1.14105i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.679058 + 1.14105i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.47 + 7.60i)T \) |
| 31 | \( 1 + (-1.63e5 - 2.78e4i)T \) |
| good | 3 | \( 1 + (23.1 + 71.2i)T + (-1.76e3 + 1.28e3i)T^{2} \) |
| 5 | \( 1 - 456.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (1.20e3 + 873. i)T + (2.54e5 + 7.83e5i)T^{2} \) |
| 11 | \( 1 + (449. + 326. i)T + (6.02e6 + 1.85e7i)T^{2} \) |
| 13 | \( 1 + (2.53e3 + 7.80e3i)T + (-5.07e7 + 3.68e7i)T^{2} \) |
| 17 | \( 1 + (-2.47e4 + 1.79e4i)T + (1.26e8 - 3.90e8i)T^{2} \) |
| 19 | \( 1 + (1.41e4 - 4.35e4i)T + (-7.23e8 - 5.25e8i)T^{2} \) |
| 23 | \( 1 + (5.32e4 - 3.86e4i)T + (1.05e9 - 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-110. + 340. i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 37 | \( 1 - 2.71e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (2.67e4 - 8.23e4i)T + (-1.57e11 - 1.14e11i)T^{2} \) |
| 43 | \( 1 + (-3.71e4 + 1.14e5i)T + (-2.19e11 - 1.59e11i)T^{2} \) |
| 47 | \( 1 + (2.32e5 + 7.14e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-2.66e5 + 1.93e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (8.94e5 + 2.75e6i)T + (-2.01e12 + 1.46e12i)T^{2} \) |
| 61 | \( 1 + 2.17e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 9.92e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (3.96e6 - 2.87e6i)T + (2.81e12 - 8.64e12i)T^{2} \) |
| 73 | \( 1 + (2.37e6 + 1.72e6i)T + (3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-3.80e6 + 2.76e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-4.90e5 + 1.51e6i)T + (-2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 + (9.91e5 + 7.20e5i)T + (1.36e13 + 4.20e13i)T^{2} \) |
| 97 | \( 1 + (8.03e6 + 5.83e6i)T + (2.49e13 + 7.68e13i)T^{2} \) |
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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
−12.88951749136537979684877457998, −12.07728110794545440831235227174, −10.27453846890466720901005907839, −9.821020885828828962961944449749, −7.70487628741751335478316601580, −6.33632521778618010406124329916, −5.64144512970734776156764353362, −3.03158652017056182490477811194, −1.60318983626600482092045061005, −0.47052567438797893143866591369,
2.74876931984500846023258684037, 4.52834746325300562203001946267, 5.82032177461154905106705699052, 6.31769834570215124234236420351, 9.014586717273298636525766384044, 9.596746672573101648401939938881, 10.38522696738414155385078144341, 12.17714529103594623181133235380, 13.34841430222169863409195301975, 14.54900716333119896294180968224
