| L(s) = 1 | + (1.73 − i)2-s + (−4.87 − 1.79i)3-s + (1.99 − 3.46i)4-s + (−9.90 − 17.1i)5-s + (−10.2 + 1.76i)6-s + (18.4 + 1.84i)7-s − 7.99i·8-s + (20.5 + 17.5i)9-s + (−34.3 − 19.8i)10-s + (−4.28 − 2.47i)11-s + (−15.9 + 13.2i)12-s + 17.7i·13-s + (33.7 − 15.2i)14-s + (17.4 + 101. i)15-s + (−8 − 13.8i)16-s + (−0.947 + 1.64i)17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.938 − 0.345i)3-s + (0.249 − 0.433i)4-s + (−0.885 − 1.53i)5-s + (−0.696 + 0.120i)6-s + (0.995 + 0.0998i)7-s − 0.353i·8-s + (0.761 + 0.648i)9-s + (−1.08 − 0.626i)10-s + (−0.117 − 0.0678i)11-s + (−0.384 + 0.319i)12-s + 0.378i·13-s + (0.644 − 0.290i)14-s + (0.301 + 1.74i)15-s + (−0.125 − 0.216i)16-s + (−0.0135 + 0.0234i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.738153 - 1.02884i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.738153 - 1.02884i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.73 + i)T \) |
| 3 | \( 1 + (4.87 + 1.79i)T \) |
| 7 | \( 1 + (-18.4 - 1.84i)T \) |
| good | 5 | \( 1 + (9.90 + 17.1i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (4.28 + 2.47i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 17.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (0.947 - 1.64i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-83.9 + 48.4i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-135. + 78.4i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 92.0iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-67.3 - 38.8i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (124. + 215. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 343.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 24.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-235. - 407. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-344. - 199. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (335. - 581. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (273. - 158. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-116. + 202. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 152. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-539. - 311. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (151. + 261. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 856.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (453. + 785. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 70.3iT - 9.12e5T^{2} \) |
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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
−15.34693247439653516844288141962, −13.67254036410809843171702437448, −12.51490874447853927046359368202, −11.84274843740368204845141486128, −10.90883772131594391175142533572, −8.855306474268862164708792042740, −7.37638031728224413032801274010, −5.29298407452059874222155678853, −4.53216331203246032270720255213, −1.10176397680589148327476835046,
3.55601823750063915409284878848, 5.17276249205504887763031714982, 6.78053219702800546221235159019, 7.79498094035497203213249681492, 10.31112279243360711403167650827, 11.32428081519054204532085476612, 11.94890517629282982176772803076, 13.80138177063983096548868457393, 15.11068880050255514369398704534, 15.41970553826490960397126000022
