L(s) = 1 | + 0.327i·2-s − 1.21·3-s + 1.89·4-s + 2.01i·5-s − 0.396i·6-s − 0.699i·7-s + 1.27i·8-s − 1.53·9-s − 0.660·10-s + 4.61i·11-s − 2.29·12-s + (3.60 − 0.0546i)13-s + 0.228·14-s − 2.44i·15-s + 3.36·16-s − 4.66·17-s + ⋯ |
L(s) = 1 | + 0.231i·2-s − 0.699·3-s + 0.946·4-s + 0.902i·5-s − 0.161i·6-s − 0.264i·7-s + 0.450i·8-s − 0.510·9-s − 0.208·10-s + 1.39i·11-s − 0.662·12-s + (0.999 − 0.0151i)13-s + 0.0611·14-s − 0.631i·15-s + 0.842·16-s − 1.13·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0151 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0151 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.861501 + 0.874656i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.861501 + 0.874656i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (-3.60 + 0.0546i)T \) |
| 31 | \( 1 - iT \) |
good | 2 | \( 1 - 0.327iT - 2T^{2} \) |
| 3 | \( 1 + 1.21T + 3T^{2} \) |
| 5 | \( 1 - 2.01iT - 5T^{2} \) |
| 7 | \( 1 + 0.699iT - 7T^{2} \) |
| 11 | \( 1 - 4.61iT - 11T^{2} \) |
| 17 | \( 1 + 4.66T + 17T^{2} \) |
| 19 | \( 1 - 2.12iT - 19T^{2} \) |
| 23 | \( 1 + 0.894T + 23T^{2} \) |
| 29 | \( 1 + 3.00T + 29T^{2} \) |
| 37 | \( 1 - 9.14iT - 37T^{2} \) |
| 41 | \( 1 - 8.88iT - 41T^{2} \) |
| 43 | \( 1 + 0.0731T + 43T^{2} \) |
| 47 | \( 1 + 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 6.80T + 53T^{2} \) |
| 59 | \( 1 + 10.8iT - 59T^{2} \) |
| 61 | \( 1 + 1.63T + 61T^{2} \) |
| 67 | \( 1 + 12.8iT - 67T^{2} \) |
| 71 | \( 1 + 7.88iT - 71T^{2} \) |
| 73 | \( 1 + 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 4.48iT - 83T^{2} \) |
| 89 | \( 1 - 5.02iT - 89T^{2} \) |
| 97 | \( 1 + 17.0iT - 97T^{2} \) |
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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
−11.39652893640043969653805959589, −10.72515186661160330102905597930, −10.06997355175509500982898638621, −8.595976318931198206007776006274, −7.46595567570110425831621494537, −6.62936942281323946540703019820, −6.13924112576351535941004108477, −4.81765386538637562379884925147, −3.28540038658611131938015598175, −1.96408967457189775180686703842,
0.875967632115337701482663071860, 2.59641348079322211643219466342, 4.00267950361931100694167825101, 5.59985500373040459701351175991, 5.96201935843958001813236880013, 7.12826077532058864483212221055, 8.586599754687066483535865806897, 8.923186500093962902362193950793, 10.58958698034810221793571833664, 11.16060430488892012409136573020