| L(s) = 1 | + (−1.34 − 1.47i)2-s + (−0.751 + 1.56i)3-s + (−0.368 + 3.98i)4-s + (1.46 + 6.40i)5-s + (3.31 − 0.991i)6-s + (−3.03 + 6.29i)7-s + (6.38 − 4.82i)8-s + (−1.87 − 2.34i)9-s + (7.49 − 10.7i)10-s + (13.6 + 10.8i)11-s + (−5.93 − 3.56i)12-s + (9.51 − 11.9i)13-s + (13.3 − 4.00i)14-s + (−11.0 − 2.53i)15-s + (−15.7 − 2.93i)16-s − 26.4·17-s + ⋯ |
| L(s) = 1 | + (−0.673 − 0.738i)2-s + (−0.250 + 0.520i)3-s + (−0.0922 + 0.995i)4-s + (0.292 + 1.28i)5-s + (0.553 − 0.165i)6-s + (−0.433 + 0.899i)7-s + (0.797 − 0.602i)8-s + (−0.207 − 0.260i)9-s + (0.749 − 1.07i)10-s + (1.24 + 0.989i)11-s + (−0.494 − 0.297i)12-s + (0.732 − 0.917i)13-s + (0.956 − 0.285i)14-s + (−0.739 − 0.168i)15-s + (−0.982 − 0.183i)16-s − 1.55·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.398724 + 0.774782i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.398724 + 0.774782i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.34 + 1.47i)T \) |
| 3 | \( 1 + (0.751 - 1.56i)T \) |
| 29 | \( 1 + (-5.22 + 28.5i)T \) |
| good | 5 | \( 1 + (-1.46 - 6.40i)T + (-22.5 + 10.8i)T^{2} \) |
| 7 | \( 1 + (3.03 - 6.29i)T + (-30.5 - 38.3i)T^{2} \) |
| 11 | \( 1 + (-13.6 - 10.8i)T + (26.9 + 117. i)T^{2} \) |
| 13 | \( 1 + (-9.51 + 11.9i)T + (-37.6 - 164. i)T^{2} \) |
| 17 | \( 1 + 26.4T + 289T^{2} \) |
| 19 | \( 1 + (-15.0 - 31.1i)T + (-225. + 282. i)T^{2} \) |
| 23 | \( 1 + (26.6 + 6.09i)T + (476. + 229. i)T^{2} \) |
| 31 | \( 1 + (-5.21 + 1.19i)T + (865. - 416. i)T^{2} \) |
| 37 | \( 1 + (-17.2 - 21.6i)T + (-304. + 1.33e3i)T^{2} \) |
| 41 | \( 1 + 44.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (23.8 + 5.43i)T + (1.66e3 + 802. i)T^{2} \) |
| 47 | \( 1 + (39.7 + 31.7i)T + (491. + 2.15e3i)T^{2} \) |
| 53 | \( 1 + (-13.1 - 57.6i)T + (-2.53e3 + 1.21e3i)T^{2} \) |
| 59 | \( 1 - 29.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + (-34.4 - 16.6i)T + (2.32e3 + 2.90e3i)T^{2} \) |
| 67 | \( 1 + (-1.00 + 0.800i)T + (998. - 4.37e3i)T^{2} \) |
| 71 | \( 1 + (-30.3 - 24.1i)T + (1.12e3 + 4.91e3i)T^{2} \) |
| 73 | \( 1 + (-19.5 + 85.6i)T + (-4.80e3 - 2.31e3i)T^{2} \) |
| 79 | \( 1 + (22.9 - 18.3i)T + (1.38e3 - 6.08e3i)T^{2} \) |
| 83 | \( 1 + (41.8 + 86.8i)T + (-4.29e3 + 5.38e3i)T^{2} \) |
| 89 | \( 1 + (33.3 + 146. i)T + (-7.13e3 + 3.43e3i)T^{2} \) |
| 97 | \( 1 + (-73.6 + 35.4i)T + (5.86e3 - 7.35e3i)T^{2} \) |
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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
−11.64059869032953959476633655682, −10.39058706693882910396035511771, −10.02493785020119916463224466181, −9.123292295338105838565490697523, −8.083059410632601623879020979905, −6.73496844321186118750173940082, −6.02238948951443360694779951842, −4.17135893106330752585441532280, −3.17431424458357281264111902521, −1.97265514164298301820546213497,
0.54073894832428434262047690583, 1.54090475413355939237640998621, 4.10784393332565095226706340754, 5.20662887977925565274201240993, 6.53056657785053071629021709859, 6.81603708498553363459006407522, 8.354039059198980008793693685972, 8.953384017239985587339844221773, 9.618707526526387726972360321301, 11.07545729484217027137000692136
