L(s) = 1 | + (1.13 − 0.848i)2-s + (0.559 − 1.92i)4-s + 0.940i·5-s + 7-s + (−0.996 − 2.64i)8-s + (0.798 + 1.06i)10-s + 5.98i·11-s + 6.59i·13-s + (1.13 − 0.848i)14-s + (−3.37 − 2.14i)16-s − 2.64·17-s + 5.83i·19-s + (1.80 + 0.526i)20-s + (5.08 + 6.77i)22-s + 2.88·23-s + ⋯ |
L(s) = 1 | + (0.799 − 0.600i)2-s + (0.279 − 0.960i)4-s + 0.420i·5-s + 0.377·7-s + (−0.352 − 0.935i)8-s + (0.252 + 0.336i)10-s + 1.80i·11-s + 1.82i·13-s + (0.302 − 0.226i)14-s + (−0.843 − 0.537i)16-s − 0.640·17-s + 1.33i·19-s + (0.403 + 0.117i)20-s + (1.08 + 1.44i)22-s + 0.601·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.546447418\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.546447418\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.13 + 0.848i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 0.940iT - 5T^{2} \) |
| 11 | \( 1 - 5.98iT - 11T^{2} \) |
| 13 | \( 1 - 6.59iT - 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 - 5.83iT - 19T^{2} \) |
| 23 | \( 1 - 2.88T + 23T^{2} \) |
| 29 | \( 1 + 3.09iT - 29T^{2} \) |
| 31 | \( 1 - 3.52T + 31T^{2} \) |
| 37 | \( 1 - 0.213iT - 37T^{2} \) |
| 41 | \( 1 + 1.63T + 41T^{2} \) |
| 43 | \( 1 + 7.16iT - 43T^{2} \) |
| 47 | \( 1 + 9.32T + 47T^{2} \) |
| 53 | \( 1 - 7.51iT - 53T^{2} \) |
| 59 | \( 1 + 11.9iT - 59T^{2} \) |
| 61 | \( 1 + 1.48iT - 61T^{2} \) |
| 67 | \( 1 + 13.0iT - 67T^{2} \) |
| 71 | \( 1 + 1.54T + 71T^{2} \) |
| 73 | \( 1 + 2.96T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 - 8.74iT - 83T^{2} \) |
| 89 | \( 1 - 7.50T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−9.668767498746022244798752878480, −9.040461082685529859232102987044, −7.72326968875456927743510121306, −6.76963621312381506815844490762, −6.42922251426175282034163872223, −4.98210427922203630794164721046, −4.51319745496531945408766962226, −3.63524681279116293016606993278, −2.24351581010220567615907674350, −1.69017686352283918033503508013,
0.77621812551885323075686197775, 2.80031094928492974397997673533, 3.31120813428015086999828418343, 4.67497543761630835555633636372, 5.24212289697588582876000739974, 6.02609648764282503601156482038, 6.87057501736891212267111295694, 7.87486725002920841370515109746, 8.527132538366501898555745502876, 8.956471110631472011734586193042