L(s) = 1 | + (−0.347 − 0.347i)2-s + (1.72 + 0.176i)3-s − 1.75i·4-s + (−1.16 − 1.90i)5-s + (−0.536 − 0.659i)6-s + (−0.707 + 0.707i)7-s + (−1.30 + 1.30i)8-s + (2.93 + 0.607i)9-s + (−0.256 + 1.06i)10-s + 2.67i·11-s + (0.310 − 3.03i)12-s + (2.14 + 2.14i)13-s + 0.490·14-s + (−1.67 − 3.49i)15-s − 2.61·16-s + (3.26 + 3.26i)17-s + ⋯ |
L(s) = 1 | + (−0.245 − 0.245i)2-s + (0.994 + 0.101i)3-s − 0.879i·4-s + (−0.522 − 0.852i)5-s + (−0.219 − 0.269i)6-s + (−0.267 + 0.267i)7-s + (−0.461 + 0.461i)8-s + (0.979 + 0.202i)9-s + (−0.0810 + 0.337i)10-s + 0.805i·11-s + (0.0895 − 0.874i)12-s + (0.596 + 0.596i)13-s + 0.131·14-s + (−0.432 − 0.901i)15-s − 0.653·16-s + (0.792 + 0.792i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.999394 - 0.477499i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.999394 - 0.477499i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.72 - 0.176i)T \) |
| 5 | \( 1 + (1.16 + 1.90i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 2 | \( 1 + (0.347 + 0.347i)T + 2iT^{2} \) |
| 11 | \( 1 - 2.67iT - 11T^{2} \) |
| 13 | \( 1 + (-2.14 - 2.14i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.26 - 3.26i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.24iT - 19T^{2} \) |
| 23 | \( 1 + (2.54 - 2.54i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 + (2.14 - 2.14i)T - 37iT^{2} \) |
| 41 | \( 1 + 11.5iT - 41T^{2} \) |
| 43 | \( 1 + (-0.759 - 0.759i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.66 + 7.66i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.43 + 4.43i)T - 53iT^{2} \) |
| 59 | \( 1 - 0.159T + 59T^{2} \) |
| 61 | \( 1 - 4.72T + 61T^{2} \) |
| 67 | \( 1 + (5.41 - 5.41i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (-4.16 - 4.16i)T + 73iT^{2} \) |
| 79 | \( 1 + 3.89iT - 79T^{2} \) |
| 83 | \( 1 + (-4.03 + 4.03i)T - 83iT^{2} \) |
| 89 | \( 1 - 3.95T + 89T^{2} \) |
| 97 | \( 1 + (1.86 - 1.86i)T - 97iT^{2} \) |
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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
−13.63660337820365609171197671895, −12.67941598058497583187607301336, −11.50414716469142426760442560121, −10.14354196050353207333152462487, −9.244318123518318236909940846262, −8.516019556209459160429086854205, −7.08719071962389929539008815808, −5.32786212587191406862732330196, −3.90661136594612864267408192239, −1.81373422795442416444879614727,
3.06180810509536809677146384878, 3.77809911175282258205806578685, 6.37827889489260480932320760699, 7.66225813772364676972655848085, 8.106865450358884966948463111391, 9.424073238928098615720663385559, 10.65168348885814778328135152447, 11.95612078147161737445790212516, 12.99614600259727023697680741029, 13.99629929881952685835538515709