L(s) = 1 | + (1.94 − 1.12i)2-s + (−0.913 − 1.47i)3-s + (1.52 − 2.64i)4-s + (0.5 − 0.866i)5-s + (−3.43 − 1.83i)6-s + (−2.43 − 1.04i)7-s − 2.38i·8-s + (−1.33 + 2.68i)9-s − 2.24i·10-s + (2.13 − 1.23i)11-s + (−5.29 + 0.170i)12-s + (0.336 + 0.194i)13-s + (−5.90 + 0.709i)14-s + (−1.73 + 0.0555i)15-s + (0.378 + 0.655i)16-s + 4.81·17-s + ⋯ |
L(s) = 1 | + (1.37 − 0.795i)2-s + (−0.527 − 0.849i)3-s + (0.764 − 1.32i)4-s + (0.223 − 0.387i)5-s + (−1.40 − 0.750i)6-s + (−0.919 − 0.393i)7-s − 0.842i·8-s + (−0.443 + 0.896i)9-s − 0.711i·10-s + (0.643 − 0.371i)11-s + (−1.52 + 0.0490i)12-s + (0.0933 + 0.0538i)13-s + (−1.57 + 0.189i)14-s + (−0.446 + 0.0143i)15-s + (0.0946 + 0.163i)16-s + 1.16·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.931845 - 1.97456i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.931845 - 1.97456i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.913 + 1.47i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.43 + 1.04i)T \) |
good | 2 | \( 1 + (-1.94 + 1.12i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.13 + 1.23i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.336 - 0.194i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.81T + 17T^{2} \) |
| 19 | \( 1 + 1.11iT - 19T^{2} \) |
| 23 | \( 1 + (1.85 + 1.06i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.02 - 2.32i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.61 - 3.24i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 + (0.476 - 0.825i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.21 + 9.03i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.81 + 4.87i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 9.72iT - 53T^{2} \) |
| 59 | \( 1 + (6.18 - 10.7i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.631 + 0.364i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.73 + 9.93i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 15.1iT - 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 + (-7.35 - 12.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.97 + 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + (3.46 - 2.00i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.74894615295419256925130375910, −10.76792749516445146453626582207, −9.860190325129948722172133017828, −8.430900394557838006497716035422, −7.01835528139239334844173661228, −6.08797825627347411692668192653, −5.33855509345817549419013469401, −4.02393217395944565249134761296, −2.84654560301654960696610651553, −1.26970940930659938881269338120,
3.11362181265542752783549358396, 3.93983559651766207906856636728, 5.10377294186547852448450149223, 6.09753131428845288344016446522, 6.49343353920677713810579871157, 7.85551013803430228752774595700, 9.542587053694655572996146607326, 9.948773310773956422032494736827, 11.41307452572542126498328815326, 12.14384960599416379104891529875