| L(s) = 1 | + (5.20 + 16.0i)2-s + (−55.0 + 11.6i)3-s + (−126. + 91.5i)4-s + (247. − 428. i)5-s + (−473. − 820. i)6-s + (−708. − 315. i)7-s + (−378. − 275. i)8-s + (892. − 397. i)9-s + (8.14e3 + 1.73e3i)10-s + (−373. − 3.55e3i)11-s + (5.86e3 − 6.51e3i)12-s + (1.53e3 + 1.70e3i)13-s + (1.36e3 − 1.29e4i)14-s + (−8.59e3 + 2.64e4i)15-s + (−3.72e3 + 1.14e4i)16-s + (3.08e3 − 2.93e4i)17-s + ⋯ |
| L(s) = 1 | + (0.460 + 1.41i)2-s + (−1.17 + 0.250i)3-s + (−0.984 + 0.715i)4-s + (0.884 − 1.53i)5-s + (−0.895 − 1.55i)6-s + (−0.780 − 0.347i)7-s + (−0.261 − 0.189i)8-s + (0.408 − 0.181i)9-s + (2.57 + 0.547i)10-s + (−0.0846 − 0.805i)11-s + (0.979 − 1.08i)12-s + (0.193 + 0.215i)13-s + (0.133 − 1.26i)14-s + (−0.657 + 2.02i)15-s + (−0.227 + 0.699i)16-s + (0.152 − 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.05125 - 0.214961i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05125 - 0.214961i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-1.52e5 - 6.54e4i)T \) |
| good | 2 | \( 1 + (-5.20 - 16.0i)T + (-103. + 75.2i)T^{2} \) |
| 3 | \( 1 + (55.0 - 11.6i)T + (1.99e3 - 889. i)T^{2} \) |
| 5 | \( 1 + (-247. + 428. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (708. + 315. i)T + (5.51e5 + 6.12e5i)T^{2} \) |
| 11 | \( 1 + (373. + 3.55e3i)T + (-1.90e7 + 4.05e6i)T^{2} \) |
| 13 | \( 1 + (-1.53e3 - 1.70e3i)T + (-6.55e6 + 6.24e7i)T^{2} \) |
| 17 | \( 1 + (-3.08e3 + 2.93e4i)T + (-4.01e8 - 8.53e7i)T^{2} \) |
| 19 | \( 1 + (-8.96e3 + 9.95e3i)T + (-9.34e7 - 8.88e8i)T^{2} \) |
| 23 | \( 1 + (4.77e4 + 3.46e4i)T + (1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (3.09e4 + 9.52e4i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 37 | \( 1 + (2.79e4 + 4.84e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (5.25e5 + 1.11e5i)T + (1.77e11 + 7.92e10i)T^{2} \) |
| 43 | \( 1 + (5.63e5 - 6.26e5i)T + (-2.84e10 - 2.70e11i)T^{2} \) |
| 47 | \( 1 + (1.07e5 - 3.31e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-1.84e6 + 8.23e5i)T + (7.86e11 - 8.72e11i)T^{2} \) |
| 59 | \( 1 + (1.27e6 - 2.70e5i)T + (2.27e12 - 1.01e12i)T^{2} \) |
| 61 | \( 1 - 1.80e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + (1.96e6 - 3.40e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-4.23e6 + 1.88e6i)T + (6.08e12 - 6.75e12i)T^{2} \) |
| 73 | \( 1 + (1.89e5 + 1.79e6i)T + (-1.08e13 + 2.29e12i)T^{2} \) |
| 79 | \( 1 + (2.04e5 - 1.94e6i)T + (-1.87e13 - 3.99e12i)T^{2} \) |
| 83 | \( 1 + (6.65e6 + 1.41e6i)T + (2.47e13 + 1.10e13i)T^{2} \) |
| 89 | \( 1 + (-7.48e6 + 5.43e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (9.64e6 - 7.00e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−16.01322299145967042970233777752, −13.87897055592467462888415916097, −13.23849667524564718173849686312, −11.77966692101611890261404115171, −9.921509858342540615193899841365, −8.494713440564351492920365006616, −6.53268811151978502017081856996, −5.58688733479458913307436313204, −4.70836004167038999267238571607, −0.50327478315714435036245565208,
1.86733534157467533055171450287, 3.36900912803944307639613041596, 5.68080063032259917930461349327, 6.77331809451368875226694297418, 9.978028506255900881394928056542, 10.47563065394956452792556293872, 11.66647031658035055835686736110, 12.62568063838361353075400679327, 13.75476901684713346980056430117, 15.16281384601678422197550364892
