L(s) = 1 | − 1.57i·2-s + (−12.8 − 8.75i)3-s + 29.5·4-s − 37.1·5-s + (−13.7 + 20.2i)6-s − 122. i·7-s − 96.6i·8-s + (89.6 + 225. i)9-s + 58.4i·10-s − 321.·11-s + (−380. − 258. i)12-s − 207.·13-s − 192.·14-s + (479. + 325. i)15-s + 793.·16-s − 1.64e3·17-s + ⋯ |
L(s) = 1 | − 0.277i·2-s + (−0.827 − 0.561i)3-s + 0.922·4-s − 0.665·5-s + (−0.156 + 0.229i)6-s − 0.942i·7-s − 0.534i·8-s + (0.368 + 0.929i)9-s + 0.184i·10-s − 0.801·11-s + (−0.763 − 0.518i)12-s − 0.340·13-s − 0.261·14-s + (0.550 + 0.373i)15-s + 0.774·16-s − 1.37·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.0821300 + 0.290284i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0821300 + 0.290284i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (12.8 + 8.75i)T \) |
| 23 | \( 1 + (2.53e3 - 114. i)T \) |
good | 2 | \( 1 + 1.57iT - 32T^{2} \) |
| 5 | \( 1 + 37.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 122. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 321.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 207.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.64e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.88e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 - 2.44e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 244.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.93e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.74e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 3.35e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.50e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 7.91e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.13e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 2.59e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 3.69e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.28e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 7.18e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.33e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 1.07e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.12e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.67e4iT - 8.58e9T^{2} \) |
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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
−12.73724175440538526055198391990, −11.96942508581739319369776670412, −10.89359521886072708207574795428, −10.28866204554034685559326446929, −7.86744900818822610412261176244, −7.17975839128468495658517875558, −5.86945294446196772370886188829, −4.02092186367666396731034276755, −1.93540279306998961845953848103, −0.13629943859688969824557755578,
2.59514621490958248657909700764, 4.65024713490708582208759881902, 5.94043935093083061912441610208, 7.09073480268786726226498414810, 8.550144246290115999739867015437, 10.07898140545330651130560359965, 11.41510568359146257611059394974, 11.68797849100888831471726096601, 13.07183357472506690832429800690, 15.10981816964075272167238151552