| L(s) = 1 | + (0.177 + 0.307i)2-s + (85.3 + 147. i)3-s + (1.02e3 − 1.77e3i)4-s − 6.27e3·5-s + (−30.2 + 52.4i)6-s + (−7.75e3 + 1.34e4i)7-s + 1.45e3·8-s + (7.39e4 − 1.28e5i)9-s + (−1.11e3 − 1.92e3i)10-s + (−4.17e5 − 7.23e5i)11-s + 3.49e5·12-s + (−1.32e6 − 1.99e5i)13-s − 5.50e3·14-s + (−5.36e5 − 9.28e5i)15-s + (−2.09e6 − 3.63e6i)16-s + (3.89e6 − 6.73e6i)17-s + ⋯ |
| L(s) = 1 | + (0.00392 + 0.00679i)2-s + (0.202 + 0.351i)3-s + (0.499 − 0.865i)4-s − 0.898·5-s + (−0.00159 + 0.00275i)6-s + (−0.174 + 0.302i)7-s + 0.0156·8-s + (0.417 − 0.723i)9-s + (−0.00352 − 0.00610i)10-s + (−0.782 − 1.35i)11-s + 0.405·12-s + (−0.988 − 0.148i)13-s − 0.00273·14-s + (−0.182 − 0.315i)15-s + (−0.499 − 0.865i)16-s + (0.664 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.664519 - 1.00754i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.664519 - 1.00754i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (1.32e6 + 1.99e5i)T \) |
| good | 2 | \( 1 + (-0.177 - 0.307i)T + (-1.02e3 + 1.77e3i)T^{2} \) |
| 3 | \( 1 + (-85.3 - 147. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + 6.27e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + (7.75e3 - 1.34e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (4.17e5 + 7.23e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 17 | \( 1 + (-3.89e6 + 6.73e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-2.93e5 + 5.07e5i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-1.37e7 - 2.37e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (-9.50e7 - 1.64e8i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 + 2.17e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + (9.80e7 + 1.69e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + (-1.93e8 - 3.35e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-6.88e8 + 1.19e9i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 - 1.32e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.20e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-1.48e9 + 2.56e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-2.21e9 + 3.83e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (7.83e9 + 1.35e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + (2.62e9 - 4.54e9i)T + (-1.15e20 - 2.00e20i)T^{2} \) |
| 73 | \( 1 - 1.12e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.13e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.31e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-2.84e10 - 4.92e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + (-5.50e10 + 9.53e10i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.16628469355944258106753532571, −15.49643321683134044504891866116, −14.24695138170378925409031486152, −12.17883124394407589988532383860, −10.82143121238974748268598912030, −9.312856277144286195198753963723, −7.31451467993472368888168018611, −5.35809990904243479343384686936, −3.12734537839031088160323031971, −0.53648961402168649349368109317,
2.32068983200142794226315968156, 4.28585714876230382007834825521, 7.25140629386288159043508807457, 7.936033176888970837971319929079, 10.38066035239296098001630129885, 12.13397983046959796892260537272, 12.96723042974441888115417321934, 15.00938772487563906811637200157, 16.21214031826734585661732872624, 17.43476085612380091846638165321
