L(s) = 1 | + (1.12 − 0.854i)2-s + (0.540 − 1.92i)4-s + (−0.763 − 0.763i)5-s + 1.33·7-s + (−1.03 − 2.63i)8-s + (−1.51 − 0.208i)10-s + (−1.95 + 1.95i)11-s + (4.18 + 4.18i)13-s + (1.50 − 1.14i)14-s + (−3.41 − 2.08i)16-s − 4.03i·17-s + (−4.26 + 4.26i)19-s + (−1.88 + 1.05i)20-s + (−0.534 + 3.88i)22-s + 8.86i·23-s + ⋯ |
L(s) = 1 | + (0.796 − 0.604i)2-s + (0.270 − 0.962i)4-s + (−0.341 − 0.341i)5-s + 0.505·7-s + (−0.366 − 0.930i)8-s + (−0.478 − 0.0658i)10-s + (−0.590 + 0.590i)11-s + (1.16 + 1.16i)13-s + (0.402 − 0.305i)14-s + (−0.854 − 0.520i)16-s − 0.978i·17-s + (−0.979 + 0.979i)19-s + (−0.421 + 0.236i)20-s + (−0.113 + 0.827i)22-s + 1.84i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.35361 - 0.838832i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35361 - 0.838832i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.12 + 0.854i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.763 + 0.763i)T + 5iT^{2} \) |
| 7 | \( 1 - 1.33T + 7T^{2} \) |
| 11 | \( 1 + (1.95 - 1.95i)T - 11iT^{2} \) |
| 13 | \( 1 + (-4.18 - 4.18i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.03iT - 17T^{2} \) |
| 19 | \( 1 + (4.26 - 4.26i)T - 19iT^{2} \) |
| 23 | \( 1 - 8.86iT - 23T^{2} \) |
| 29 | \( 1 + (1.23 - 1.23i)T - 29iT^{2} \) |
| 31 | \( 1 + 2.87iT - 31T^{2} \) |
| 37 | \( 1 + (-0.434 + 0.434i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.81T + 41T^{2} \) |
| 43 | \( 1 + (5.49 + 5.49i)T + 43iT^{2} \) |
| 47 | \( 1 - 3.20T + 47T^{2} \) |
| 53 | \( 1 + (4.06 + 4.06i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.71 + 4.71i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3.26 - 3.26i)T + 61iT^{2} \) |
| 67 | \( 1 + (5.44 - 5.44i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.76iT - 71T^{2} \) |
| 73 | \( 1 + 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 11.1iT - 79T^{2} \) |
| 83 | \( 1 + (9.73 + 9.73i)T + 83iT^{2} \) |
| 89 | \( 1 - 1.64T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
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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
−12.93284940847541405399189307670, −11.83692878391796435240340789853, −11.24916309364268471082011327030, −10.06925890591722147607370683451, −8.923885710538010418543982207923, −7.51744553197712260346154714656, −6.10645358015957432775583308745, −4.82888994961844820546655956003, −3.79017462950688121524413506771, −1.85786522019020539428296435516,
2.91351473777109356366258258133, 4.26709991156210722201910819356, 5.63090772232286782571788032644, 6.64621052386853177098683182746, 8.059316769708845201460744944031, 8.548272960770936051295246193254, 10.75248620400562165791615774241, 11.13660906649380794199119324923, 12.69550370210802336437058882068, 13.17518285154771886950654893031