L(s) = 1 | + (−0.190 + 0.109i)2-s + (0.800 + 1.38i)3-s + (−0.975 + 1.69i)4-s − i·5-s + (−0.304 − 0.175i)6-s + (−0.287 − 0.166i)7-s − 0.868i·8-s + (0.219 − 0.380i)9-s + (0.109 + 0.190i)10-s + (4.65 − 2.68i)11-s − 3.12·12-s + (−3.55 − 0.619i)13-s + 0.0729·14-s + (1.38 − 0.800i)15-s + (−1.85 − 3.21i)16-s + (−2.53 + 4.38i)17-s + ⋯ |
L(s) = 1 | + (−0.134 + 0.0776i)2-s + (0.461 + 0.800i)3-s + (−0.487 + 0.845i)4-s − 0.447i·5-s + (−0.124 − 0.0717i)6-s + (−0.108 − 0.0627i)7-s − 0.306i·8-s + (0.0732 − 0.126i)9-s + (0.0347 + 0.0601i)10-s + (1.40 − 0.809i)11-s − 0.901·12-s + (−0.985 − 0.171i)13-s + 0.0195·14-s + (0.357 − 0.206i)15-s + (−0.464 − 0.803i)16-s + (−0.614 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.801290 + 0.368313i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.801290 + 0.368313i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + iT \) |
| 13 | \( 1 + (3.55 + 0.619i)T \) |
good | 2 | \( 1 + (0.190 - 0.109i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.800 - 1.38i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (0.287 + 0.166i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.65 + 2.68i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.53 - 4.38i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.96 + 1.13i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.41 + 2.45i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.45 - 2.51i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.46iT - 31T^{2} \) |
| 37 | \( 1 + (5.17 - 2.98i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.23 + 1.86i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.53 - 4.38i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 8.34iT - 47T^{2} \) |
| 53 | \( 1 + 1.56T + 53T^{2} \) |
| 59 | \( 1 + (-2.34 - 1.35i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.05 - 12.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.94 + 5.16i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (11.0 + 6.39i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9.68iT - 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 + 4.26iT - 83T^{2} \) |
| 89 | \( 1 + (2.79 - 1.61i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.17 - 1.25i)T + (48.5 + 84.0i)T^{2} \) |
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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
−15.03475765989790546639160388568, −14.08034281759538355325928148957, −12.80949798648276166353144607573, −11.90231065390357000809432293997, −10.23487566878325788359240402754, −9.018755820104747997268103713093, −8.510132644312566629443595730688, −6.71788102904751183275109623967, −4.55811518716557162412556913308, −3.51165273711499983201429723159,
2.03773300000244614306454951812, 4.56804986037642860765581972549, 6.43985310246550613953770658146, 7.50198416305323760691918524689, 9.126067418043696733670669210564, 9.958380409594795905148953487384, 11.44146993080559789274857062193, 12.65687830642304021895992346384, 13.93914012754837913755165049793, 14.41937148184295279441568938881