| L(s) = 1 | + (−2.35 − 3.23i)2-s + (25.8 + 8.40i)3-s + (−4.94 + 15.2i)4-s + (52.0 − 20.4i)5-s + (−33.6 − 103. i)6-s + 51.8i·7-s + (60.8 − 19.7i)8-s + (401. + 291. i)9-s + (−188. − 120. i)10-s + (−441. + 320. i)11-s + (−255. + 352. i)12-s + (144. − 198. i)13-s + (167. − 121. i)14-s + (1.51e3 − 91.6i)15-s + (−207. − 150. i)16-s + (1.80e3 − 587. i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (1.65 + 0.539i)3-s + (−0.154 + 0.475i)4-s + (0.930 − 0.365i)5-s + (−0.381 − 1.17i)6-s + 0.399i·7-s + (0.336 − 0.109i)8-s + (1.65 + 1.20i)9-s + (−0.596 − 0.380i)10-s + (−1.10 + 0.799i)11-s + (−0.512 + 0.705i)12-s + (0.236 − 0.325i)13-s + (0.228 − 0.166i)14-s + (1.74 − 0.105i)15-s + (−0.202 − 0.146i)16-s + (1.51 − 0.492i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0678i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.50075 - 0.0849407i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.50075 - 0.0849407i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.35 + 3.23i)T \) |
| 5 | \( 1 + (-52.0 + 20.4i)T \) |
| good | 3 | \( 1 + (-25.8 - 8.40i)T + (196. + 142. i)T^{2} \) |
| 7 | \( 1 - 51.8iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (441. - 320. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-144. + 198. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-1.80e3 + 587. i)T + (1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (261. + 803. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (1.02e3 + 1.40e3i)T + (-1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (2.24e3 - 6.90e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (2.43e3 + 7.50e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (6.37e3 - 8.77e3i)T + (-2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (3.54e3 + 2.57e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 5.77e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (1.27e4 + 4.13e3i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (2.52e4 + 8.21e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.75e4 + 1.27e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (9.28e3 - 6.74e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (4.11e4 - 1.33e4i)T + (1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-1.05e4 + 3.23e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-3.72e4 - 5.13e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.68e4 + 8.24e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (5.35e4 - 1.73e4i)T + (3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (5.91e4 - 4.29e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-1.22e5 - 3.96e4i)T + (6.94e9 + 5.04e9i)T^{2} \) |
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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
−14.44687172084838980901465249109, −13.39276555313352232722279052901, −12.53662053451520781333238390446, −10.38751566139678924609047197816, −9.692901277308007401046624227943, −8.733851134083911109070087104533, −7.65561010651625364679702403868, −5.02700019194702834015200020298, −3.12084539742165696537323727904, −1.97048689577880955010747396522,
1.65090561055304345762223972218, 3.28222345572865424449615050661, 5.89290118230205749677610686619, 7.43745688556443731516655273431, 8.271204931018872514560222472829, 9.504165170700451289022249840419, 10.47965791842804623409225445390, 12.83652586340516249135649507335, 13.88402915359966289411312104139, 14.25346938057944414451394133755
