L(s) = 1 | + (0.492 + 1.66i)3-s − 5-s − 1.32i·7-s + (−2.51 + 1.63i)9-s − 5.02i·11-s − 6.73i·13-s + (−0.492 − 1.66i)15-s + 5.43i·17-s − 6.73·19-s + (2.19 − 0.651i)21-s − 5.75·23-s + 25-s + (−3.95 − 3.36i)27-s − 1.61·29-s − 7.02i·31-s + ⋯ |
L(s) = 1 | + (0.284 + 0.958i)3-s − 0.447·5-s − 0.499i·7-s + (−0.838 + 0.545i)9-s − 1.51i·11-s − 1.86i·13-s + (−0.127 − 0.428i)15-s + 1.31i·17-s − 1.54·19-s + (0.478 − 0.142i)21-s − 1.19·23-s + 0.200·25-s + (−0.761 − 0.648i)27-s − 0.299·29-s − 1.26i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0267 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0267 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.597128 - 0.581362i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.597128 - 0.581362i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.492 - 1.66i)T \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + 1.32iT - 7T^{2} \) |
| 11 | \( 1 + 5.02iT - 11T^{2} \) |
| 13 | \( 1 + 6.73iT - 13T^{2} \) |
| 17 | \( 1 - 5.43iT - 17T^{2} \) |
| 19 | \( 1 + 6.73T + 19T^{2} \) |
| 23 | \( 1 + 5.75T + 23T^{2} \) |
| 29 | \( 1 + 1.61T + 29T^{2} \) |
| 31 | \( 1 + 7.02iT - 31T^{2} \) |
| 37 | \( 1 + 4.76iT - 37T^{2} \) |
| 41 | \( 1 + 3.27iT - 41T^{2} \) |
| 43 | \( 1 - 2.95T + 43T^{2} \) |
| 47 | \( 1 - 0.795T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 1.61iT - 59T^{2} \) |
| 61 | \( 1 - 5.24iT - 61T^{2} \) |
| 67 | \( 1 + 5.56T + 67T^{2} \) |
| 71 | \( 1 + 4.95T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 - 6.44iT - 79T^{2} \) |
| 83 | \( 1 + 8.67iT - 83T^{2} \) |
| 89 | \( 1 - 16.4iT - 89T^{2} \) |
| 97 | \( 1 - 8.64T + 97T^{2} \) |
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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
−10.08176735054384969324889582318, −8.770143404028469770968350094709, −8.297076796541568920711047924912, −7.65306641552289297584277690349, −6.00203525510236094913704264958, −5.64149964111957311361365094319, −4.08757034199464053081805774402, −3.74738335313974300527719470341, −2.54994475611538998064779711273, −0.35419094117649581559268952278,
1.77514820813019085188139069986, 2.49241315033723664306535747228, 4.03607916747701819084033727627, 4.86672930371845481441394534579, 6.27747873202660861569051735168, 6.93416813644304600810588457004, 7.54138705771961457923649202118, 8.641999577792661643714757698662, 9.158188203539845076309086826451, 10.10087707180344808044751411633