| L(s) = 1 | + (−1.83 + 1.70i)2-s + (−1.89 + 0.285i)3-s + (−0.129 + 1.73i)4-s + (−0.225 + 0.575i)5-s + (2.98 − 3.74i)6-s + (−16.4 − 8.59i)7-s + (−15.1 − 19.0i)8-s + (−22.3 + 6.87i)9-s + (−0.564 − 1.43i)10-s + (−0.936 − 0.288i)11-s + (−0.248 − 3.31i)12-s + (−3.83 − 16.7i)13-s + (44.7 − 12.1i)14-s + (0.263 − 1.15i)15-s + (46.5 + 7.01i)16-s + (−75.8 + 51.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.648 + 0.601i)2-s + (−0.364 + 0.0548i)3-s + (−0.0162 + 0.216i)4-s + (−0.0201 + 0.0514i)5-s + (0.203 − 0.254i)6-s + (−0.885 − 0.464i)7-s + (−0.671 − 0.841i)8-s + (−0.825 + 0.254i)9-s + (−0.0178 − 0.0455i)10-s + (−0.0256 − 0.00792i)11-s + (−0.00597 − 0.0797i)12-s + (−0.0817 − 0.358i)13-s + (0.853 − 0.232i)14-s + (0.00452 − 0.0198i)15-s + (0.727 + 0.109i)16-s + (−1.08 + 0.737i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0434081 - 0.201646i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0434081 - 0.201646i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (16.4 + 8.59i)T \) |
| good | 2 | \( 1 + (1.83 - 1.70i)T + (0.597 - 7.97i)T^{2} \) |
| 3 | \( 1 + (1.89 - 0.285i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (0.225 - 0.575i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (0.936 + 0.288i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (3.83 + 16.7i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (75.8 - 51.7i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (-64.5 - 111. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-6.96 - 4.74i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (244. + 117. i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (86.4 - 149. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-2.36 - 31.5i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (116. + 145. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-55.9 + 70.1i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (238. - 221. i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (-34.8 + 464. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (-138. - 353. i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (-16.1 - 215. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (-197. + 342. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-421. + 202. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (439. + 408. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (1.61 + 2.80i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (118. - 519. i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (1.09e3 - 336. i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 - 224.T + 9.12e5T^{2} \) |
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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
−16.14771016088920568680252017056, −14.92196577646348878791886860016, −13.39245722787594832407721251639, −12.36129367721967500516096266497, −10.91204590351781990966262438006, −9.611819993215053067478016607551, −8.376101187635175191749870416438, −7.10714350340127477777088431242, −5.84261348741391777945435849278, −3.50490027602114768388754010819,
0.19064241085713251430297971178, 2.70334185621093675500959253769, 5.33565511337876787145698927265, 6.70412875866848640456186508946, 8.877049091731228159036407239464, 9.493884667305114779867248818181, 11.03867566719148760444749181162, 11.72245214206949394248427273653, 13.12420583959882625239261625859, 14.52148392857907283044742431174
