| L(s) = 1 | + (1.73 + i)2-s + (−0.0354 − 5.19i)3-s + (1.99 + 3.46i)4-s + (5.27 − 9.13i)5-s + (5.13 − 9.03i)6-s + (17.7 + 5.44i)7-s + 7.99i·8-s + (−26.9 + 0.368i)9-s + (18.2 − 10.5i)10-s + (−26.6 + 15.4i)11-s + (17.9 − 10.5i)12-s + 19.8i·13-s + (25.2 + 27.1i)14-s + (−47.6 − 27.0i)15-s + (−8 + 13.8i)16-s + (46.3 + 80.2i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.00682 − 0.999i)3-s + (0.249 + 0.433i)4-s + (0.471 − 0.816i)5-s + (0.349 − 0.614i)6-s + (0.955 + 0.293i)7-s + 0.353i·8-s + (−0.999 + 0.0136i)9-s + (0.577 − 0.333i)10-s + (−0.731 + 0.422i)11-s + (0.431 − 0.252i)12-s + 0.423i·13-s + (0.481 + 0.517i)14-s + (−0.820 − 0.466i)15-s + (−0.125 + 0.216i)16-s + (0.661 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.83824 - 0.390010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83824 - 0.390010i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (0.0354 + 5.19i)T \) |
| 7 | \( 1 + (-17.7 - 5.44i)T \) |
| good | 5 | \( 1 + (-5.27 + 9.13i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (26.6 - 15.4i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 19.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-46.3 - 80.2i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (118. + 68.6i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (37.6 + 21.7i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 134. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-144. + 83.6i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-191. + 332. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 107.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-120. + 209. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (432. - 249. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (366. + 634. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (265. + 153. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (280. + 485. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 74.2iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-141. + 81.6i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (437. - 757. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 406.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-526. + 911. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 243. iT - 9.12e5T^{2} \) |
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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
−15.17648648750828974949120482401, −14.19391316418750012360440753891, −12.98338206675768412827077668516, −12.41263086843918423103579029072, −10.98130167807522599160692358401, −8.773425784753945180249404878314, −7.76513320769949970736769531667, −6.13340063726342170686393907331, −4.84572969788875609201450017213, −1.98082379158613252076274990060,
2.84514711539733920434908448416, 4.62591274589536145332909234827, 5.99815392401085525427614582709, 8.095798063112557242878682081070, 10.03222103489159942820992607459, 10.69412920404838841349077719668, 11.78324073989004960734408478777, 13.62236499858669266435999108889, 14.45560024910314555488299720255, 15.27474698719951094699315171548
