L(s) = 1 | + (1.53 + 0.5i)2-s + (0.587 + 0.809i)3-s + (0.5 + 0.363i)4-s + (0.5 + 1.53i)6-s + (3.07 − 4.23i)7-s + (−1.31 − 1.80i)8-s + (−0.309 + 0.951i)9-s + (−1.23 + 3.07i)11-s + 0.618i·12-s + (3.07 + i)13-s + (6.85 − 4.97i)14-s + (−1.50 − 4.61i)16-s + (1.90 − 0.618i)17-s + (−0.951 + 1.30i)18-s + (4.04 − 2.93i)19-s + ⋯ |
L(s) = 1 | + (1.08 + 0.353i)2-s + (0.339 + 0.467i)3-s + (0.250 + 0.181i)4-s + (0.204 + 0.628i)6-s + (1.16 − 1.60i)7-s + (−0.464 − 0.639i)8-s + (−0.103 + 0.317i)9-s + (−0.372 + 0.927i)11-s + 0.178i·12-s + (0.853 + 0.277i)13-s + (1.83 − 1.33i)14-s + (−0.375 − 1.15i)16-s + (0.461 − 0.149i)17-s + (−0.224 + 0.308i)18-s + (0.928 − 0.674i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.15959 + 0.225796i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.15959 + 0.225796i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (1.23 - 3.07i)T \) |
good | 2 | \( 1 + (-1.53 - 0.5i)T + (1.61 + 1.17i)T^{2} \) |
| 7 | \( 1 + (-3.07 + 4.23i)T + (-2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-3.07 - i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.90 + 0.618i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-4.04 + 2.93i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 4.61iT - 23T^{2} \) |
| 29 | \( 1 + (-0.690 - 0.502i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.16 + 6.65i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.865 - 1.19i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (9.28 - 6.74i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 9.09iT - 43T^{2} \) |
| 47 | \( 1 + (3.57 + 4.92i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (5.11 + 1.66i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.42 - 3.94i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.16 - 6.65i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 11.6iT - 67T^{2} \) |
| 71 | \( 1 + (2.47 + 7.60i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.66 - 3.66i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (0.954 - 2.93i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (7.33 - 2.38i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 4.14T + 89T^{2} \) |
| 97 | \( 1 + (2.40 + 0.781i)T + (78.4 + 57.0i)T^{2} \) |
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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
−10.07651280121155459002629407858, −9.700577619779828408142991872687, −8.343839256170575462074065475983, −7.49601999284845669842209864238, −6.83712685628758787282824524871, −5.48459527451703659642656939506, −4.69774550339158896269572628264, −4.15369942196862917289729335932, −3.19780678861399921008338114100, −1.32564395579228483047443811847,
1.69108236050849943620575193896, 2.80338769844465077623753462629, 3.57583232983817123478694661696, 5.04315816538455915695870002478, 5.52551058224289484041078094811, 6.33782599026812970134025220410, 7.980943965385744521232553965046, 8.444131905575754138877777093662, 8.993290511148648499008669884853, 10.54107880503562084227121999056