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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24106 + 0.0166022i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24106 + 0.0166022i\) |
\(L(\frac{3}{2})\) |
|
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
−10.21094601715172106932393772829, −9.043963100940011747835070284867, −8.152193943145729485321899696285, −7.47591568634270147745145009375, −6.84270422788753516988036088337, −5.56735829898205818406547675809, −5.12782204398199375243021855362, −3.53231073690860092856499855370, −2.45500238436513593033763886399, −1.10850982366940730972033120221,
0.75804052334874974227723876057, 2.93366950448713220454952376883, 3.76567373738903166596912433032, 4.68198177966655569521827502947, 5.52231288318084650557855170243, 6.62001299834210009081116984119, 7.42929463431068348436939699925, 8.615696568836205636899603288157, 9.268776026640275947045373542758, 9.895394937654590442482575307364