| L(s) = 1 | + (−0.347 − 1.96i)2-s + (−0.0459 + 0.260i)3-s + (−3.75 + 1.36i)4-s + (12.6 + 10.6i)5-s + 0.529·6-s + (0.748 + 0.627i)7-s + (4 + 6.92i)8-s + (25.3 + 9.21i)9-s + (16.4 − 28.5i)10-s + (−9.26 − 16.0i)11-s + (−0.183 − 1.04i)12-s + (47.8 − 17.4i)13-s + (0.976 − 1.69i)14-s + (−3.34 + 2.80i)15-s + (12.2 − 10.2i)16-s + (5.74 + 2.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.00884 + 0.0501i)3-s + (−0.469 + 0.171i)4-s + (1.13 + 0.948i)5-s + 0.0359·6-s + (0.0403 + 0.0338i)7-s + (0.176 + 0.306i)8-s + (0.937 + 0.341i)9-s + (0.521 − 0.903i)10-s + (−0.253 − 0.439i)11-s + (−0.00442 − 0.0250i)12-s + (1.02 − 0.371i)13-s + (0.0186 − 0.0322i)14-s + (−0.0575 + 0.0482i)15-s + (0.191 − 0.160i)16-s + (0.0819 + 0.0298i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.62561 - 0.188716i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.62561 - 0.188716i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.347 + 1.96i)T \) |
| 37 | \( 1 + (217. - 57.1i)T \) |
| good | 3 | \( 1 + (0.0459 - 0.260i)T + (-25.3 - 9.23i)T^{2} \) |
| 5 | \( 1 + (-12.6 - 10.6i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-0.748 - 0.627i)T + (59.5 + 337. i)T^{2} \) |
| 11 | \( 1 + (9.26 + 16.0i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-47.8 + 17.4i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-5.74 - 2.08i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 19 | \( 1 + (-0.257 + 1.45i)T + (-6.44e3 - 2.34e3i)T^{2} \) |
| 23 | \( 1 + (42.1 - 73.0i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-6.91 - 11.9i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 5.83T + 2.97e4T^{2} \) |
| 41 | \( 1 + (194. - 70.7i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + 524.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-222. + 385. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-487. + 409. i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-327. + 275. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (334. - 121. i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-303. - 254. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-23.0 + 130. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + 619.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (533. + 448. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (1.07e3 + 390. i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (-770. + 646. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (568. - 985. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.59385790320835073162311429001, −13.28363547507299449275937228805, −11.63132504224098243050416474373, −10.43408300903770948926895264048, −10.00710095858161581534423922401, −8.478419442037638378756558892723, −6.85554509446527193385785963204, −5.45176256956952508478012123988, −3.45024229493792744695617117670, −1.80646095725382573669499137892,
1.45309271504594418619559017216, 4.39144940479984577182348685126, 5.70816106606832750826846392899, 6.88427556703687838043891909738, 8.457598402343883456987990969243, 9.434896495451637077323859211762, 10.37974690429760440174587628174, 12.30273312454326308144151313501, 13.21937460775176595052166755290, 13.97334829101779871834981977470
