L(s) = 1 | + (41.7 − 48.5i)2-s + (193. − 193. i)3-s + (−612. − 4.04e3i)4-s + (−1.03e4 + 1.03e4i)5-s + (−1.31e3 − 1.74e4i)6-s − 1.08e5·7-s + (−2.22e5 − 1.39e5i)8-s + 4.56e5i·9-s + (7.02e4 + 9.34e5i)10-s + (−1.04e6 − 1.04e6i)11-s + (−9.01e5 − 6.64e5i)12-s + (−1.27e6 − 1.27e6i)13-s + (−4.52e6 + 5.26e6i)14-s + 4.00e6i·15-s + (−1.60e7 + 4.96e6i)16-s + 1.95e7·17-s + ⋯ |
L(s) = 1 | + (0.652 − 0.758i)2-s + (0.265 − 0.265i)3-s + (−0.149 − 0.988i)4-s + (−0.662 + 0.662i)5-s + (−0.0281 − 0.373i)6-s − 0.921·7-s + (−0.847 − 0.531i)8-s + 0.859i·9-s + (0.0702 + 0.934i)10-s + (−0.590 − 0.590i)11-s + (−0.301 − 0.222i)12-s + (−0.264 − 0.264i)13-s + (−0.601 + 0.698i)14-s + 0.351i·15-s + (−0.955 + 0.295i)16-s + 0.808·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(0.106038 + 0.225876i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.106038 + 0.225876i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-41.7 + 48.5i)T \) |
good | 3 | \( 1 + (-193. + 193. i)T - 5.31e5iT^{2} \) |
| 5 | \( 1 + (1.03e4 - 1.03e4i)T - 2.44e8iT^{2} \) |
| 7 | \( 1 + 1.08e5T + 1.38e10T^{2} \) |
| 11 | \( 1 + (1.04e6 + 1.04e6i)T + 3.13e12iT^{2} \) |
| 13 | \( 1 + (1.27e6 + 1.27e6i)T + 2.32e13iT^{2} \) |
| 17 | \( 1 - 1.95e7T + 5.82e14T^{2} \) |
| 19 | \( 1 + (2.72e7 - 2.72e7i)T - 2.21e15iT^{2} \) |
| 23 | \( 1 + 1.76e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (3.27e8 + 3.27e8i)T + 3.53e17iT^{2} \) |
| 31 | \( 1 + 1.40e9iT - 7.87e17T^{2} \) |
| 37 | \( 1 + (-3.08e9 + 3.08e9i)T - 6.58e18iT^{2} \) |
| 41 | \( 1 + 1.10e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + (4.13e9 + 4.13e9i)T + 3.99e19iT^{2} \) |
| 47 | \( 1 + 7.39e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 + (2.73e10 - 2.73e10i)T - 4.91e20iT^{2} \) |
| 59 | \( 1 + (-4.52e9 - 4.52e9i)T + 1.77e21iT^{2} \) |
| 61 | \( 1 + (-2.45e9 - 2.45e9i)T + 2.65e21iT^{2} \) |
| 67 | \( 1 + (2.12e9 - 2.12e9i)T - 8.18e21iT^{2} \) |
| 71 | \( 1 - 5.19e10T + 1.64e22T^{2} \) |
| 73 | \( 1 - 2.15e11iT - 2.29e22T^{2} \) |
| 79 | \( 1 - 3.83e10iT - 5.90e22T^{2} \) |
| 83 | \( 1 + (-3.95e11 + 3.95e11i)T - 1.06e23iT^{2} \) |
| 89 | \( 1 - 7.22e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + 1.01e12T + 6.93e23T^{2} \) |
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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
−15.01957787423387153830618311346, −13.72245379660269972690997169280, −12.62919952759020062749342348915, −11.14700583490512884244608134886, −9.964427772105536416637700926032, −7.76247969518470639325254233723, −5.85598680039794639302993500228, −3.72761870438839257011962777326, −2.42459839448482057215573996888, −0.07441663744463600016696007147,
3.26542536854695070391282484714, 4.65040796180522670387634452968, 6.52691604536041484790682008318, 8.131184836235198192487687792549, 9.585817192630555973755078012716, 12.08964716791311844286813044268, 12.89473611880678801085583688351, 14.57116121723174566773114379652, 15.69680890752411944948996875461, 16.48114016344170639063822865050