L(s) = 1 | − 3.74e10i·2-s + (−7.56e15 − 4.94e16i)3-s − 2.23e20·4-s + 1.07e24i·5-s + (−1.85e27 + 2.83e26i)6-s + 6.73e29·7-s − 3.58e31i·8-s + (−2.38e33 + 7.47e32i)9-s + 4.02e34·10-s + 9.56e35i·11-s + (1.68e36 + 1.10e37i)12-s + 9.40e38·13-s − 2.52e40i·14-s + (5.31e40 − 8.12e39i)15-s − 1.60e42·16-s + 1.38e43i·17-s + ⋯ |
L(s) = 1 | − 1.09i·2-s + (−0.151 − 0.988i)3-s − 0.189·4-s + 0.369i·5-s + (−1.07 + 0.164i)6-s + 1.77·7-s − 0.884i·8-s + (−0.954 + 0.298i)9-s + 0.402·10-s + 0.340i·11-s + (0.0285 + 0.186i)12-s + 0.966·13-s − 1.93i·14-s + (0.364 − 0.0557i)15-s − 1.15·16-s + 1.19i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(71-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{71}{2})\) |
\(\approx\) |
\(3.019741923\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.019741923\) |
\(L(36)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (7.56e15 + 4.94e16i)T \) |
good | 2 | \( 1 + 3.74e10iT - 1.18e21T^{2} \) |
| 5 | \( 1 - 1.07e24iT - 8.47e48T^{2} \) |
| 7 | \( 1 - 6.73e29T + 1.43e59T^{2} \) |
| 11 | \( 1 - 9.56e35iT - 7.89e72T^{2} \) |
| 13 | \( 1 - 9.40e38T + 9.46e77T^{2} \) |
| 17 | \( 1 - 1.38e43iT - 1.35e86T^{2} \) |
| 19 | \( 1 + 4.21e44T + 3.25e89T^{2} \) |
| 23 | \( 1 - 7.41e47iT - 2.09e95T^{2} \) |
| 29 | \( 1 - 6.56e50iT - 2.33e102T^{2} \) |
| 31 | \( 1 - 2.15e52T + 2.48e104T^{2} \) |
| 37 | \( 1 - 4.93e54T + 5.94e109T^{2} \) |
| 41 | \( 1 + 5.23e55iT - 7.85e112T^{2} \) |
| 43 | \( 1 - 1.48e57T + 2.20e114T^{2} \) |
| 47 | \( 1 - 1.27e58iT - 1.11e117T^{2} \) |
| 53 | \( 1 - 3.94e60iT - 5.00e120T^{2} \) |
| 59 | \( 1 - 9.57e61iT - 9.11e123T^{2} \) |
| 61 | \( 1 + 2.37e61T + 9.39e124T^{2} \) |
| 67 | \( 1 + 8.71e63T + 6.68e127T^{2} \) |
| 71 | \( 1 - 4.27e64iT - 3.87e129T^{2} \) |
| 73 | \( 1 - 1.50e65T + 2.70e130T^{2} \) |
| 79 | \( 1 + 2.81e66T + 6.82e132T^{2} \) |
| 83 | \( 1 - 1.40e66iT - 2.16e134T^{2} \) |
| 89 | \( 1 + 8.71e67iT - 2.86e136T^{2} \) |
| 97 | \( 1 + 5.94e69T + 1.18e139T^{2} \) |
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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
−12.42745087170852870526321243169, −11.31282757176774526252002124447, −10.71277841713680011779913197582, −8.562080491860048862983991298328, −7.38814172544820081603848536385, −5.97142487375902706706662540662, −4.25273968405749688605031349324, −2.71018966995499253192309596806, −1.56145889530364022457106557457, −1.19325894102919511950457936461,
0.74980424145027776460087582778, 2.45561123325711016367799402976, 4.45059907015026820009746604349, 5.09788506412877941875475666140, 6.35805213740660099744861622096, 8.123883035045591501259519971097, 8.735776127047022254453309874162, 10.78150939568833743957437502322, 11.60206000272563633955122060111, 14.05344086411252730807444091693