| L(s) = 1 | − 21.6·2-s + (−19.2 + 33.4i)3-s + 342.·4-s + (52.6 − 91.1i)5-s + (418. − 724. i)6-s + (19.7 + 34.2i)7-s − 4.65e3·8-s + (349. + 604. i)9-s + (−1.14e3 + 1.97e3i)10-s − 5.17e3·11-s + (−6.60e3 + 1.14e4i)12-s + (−1.32e3 − 2.29e3i)13-s + (−429. − 743. i)14-s + (2.03e3 + 3.51e3i)15-s + 5.71e4·16-s + (−1.00e4 − 1.73e4i)17-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + (−0.412 + 0.714i)3-s + 2.67·4-s + (0.188 − 0.326i)5-s + (0.791 − 1.37i)6-s + (0.0218 + 0.0377i)7-s − 3.21·8-s + (0.159 + 0.276i)9-s + (−0.361 + 0.625i)10-s − 1.17·11-s + (−1.10 + 1.91i)12-s + (−0.167 − 0.289i)13-s + (−0.0418 − 0.0724i)14-s + (0.155 + 0.269i)15-s + 3.48·16-s + (−0.495 − 0.858i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.455321 - 0.122796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.455321 - 0.122796i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-5.09e5 + 1.11e5i)T \) |
| good | 2 | \( 1 + 21.6T + 128T^{2} \) |
| 3 | \( 1 + (19.2 - 33.4i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-52.6 + 91.1i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-19.7 - 34.2i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + 5.17e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (1.32e3 + 2.29e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.00e4 + 1.73e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (4.09e3 - 7.09e3i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-3.92e4 + 6.80e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-6.60e4 - 1.14e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-3.02e4 + 5.23e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-1.95e5 + 3.38e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 2.11e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 9.81e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-5.89e5 + 1.02e6i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 - 2.92e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-6.26e5 - 1.08e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.80e6 + 3.13e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.64e6 + 4.58e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-2.73e6 - 4.72e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.15e6 - 3.72e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (7.74e5 - 1.34e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-1.29e6 + 2.24e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 5.65e6T + 8.07e13T^{2} \) |
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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
−15.05932916592475532100476524108, −12.80602809608507817733135902906, −11.19413057208692381698350097733, −10.49224303428467118001276972617, −9.522630593768371828998633048072, −8.344852194629248462394599151118, −7.09434518629929679778809415625, −5.27949922001980482055621138234, −2.42786637339190154718516390319, −0.48990531594695886761561419640,
0.970831420480741928785593429285, 2.45586251073243099039123937993, 6.16153443035101070514715864415, 7.16748536606991768030302298914, 8.248855914256040001048785557290, 9.652502712670131825162894226498, 10.69057063875113593310154815097, 11.71693219500892256964280885517, 13.00960987601278534523730776885, 15.09358319354313376177678530893
