L(s) = 1 | + (9.88 − 30.4i)2-s + (106. − 77.1i)3-s + (−828. − 601. i)4-s + (1.48e3 + 4.57e3i)5-s + (−1.29e3 − 3.99e3i)6-s + (9.58e3 + 6.96e3i)7-s + (−2.65e4 + 1.92e4i)8-s + (−4.94e4 + 1.52e5i)9-s + 1.54e5·10-s + (3.99e5 + 3.55e5i)11-s − 1.34e5·12-s + (−2.16e4 + 6.65e4i)13-s + (3.06e5 − 2.22e5i)14-s + (5.11e5 + 3.71e5i)15-s + (3.24e5 + 9.97e5i)16-s + (7.02e5 + 2.16e6i)17-s + ⋯ |
L(s) = 1 | + (0.218 − 0.672i)2-s + (0.252 − 0.183i)3-s + (−0.404 − 0.293i)4-s + (0.212 + 0.655i)5-s + (−0.0681 − 0.209i)6-s + (0.215 + 0.156i)7-s + (−0.286 + 0.207i)8-s + (−0.278 + 0.858i)9-s + 0.487·10-s + (0.747 + 0.664i)11-s − 0.155·12-s + (−0.0161 + 0.0497i)13-s + (0.152 − 0.110i)14-s + (0.173 + 0.126i)15-s + (0.0772 + 0.237i)16-s + (0.119 + 0.369i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(2.17958 + 0.286071i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.17958 + 0.286071i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-9.88 + 30.4i)T \) |
| 11 | \( 1 + (-3.99e5 - 3.55e5i)T \) |
good | 3 | \( 1 + (-106. + 77.1i)T + (5.47e4 - 1.68e5i)T^{2} \) |
| 5 | \( 1 + (-1.48e3 - 4.57e3i)T + (-3.95e7 + 2.87e7i)T^{2} \) |
| 7 | \( 1 + (-9.58e3 - 6.96e3i)T + (6.11e8 + 1.88e9i)T^{2} \) |
| 13 | \( 1 + (2.16e4 - 6.65e4i)T + (-1.44e12 - 1.05e12i)T^{2} \) |
| 17 | \( 1 + (-7.02e5 - 2.16e6i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (-1.67e6 + 1.21e6i)T + (3.59e13 - 1.10e14i)T^{2} \) |
| 23 | \( 1 - 5.38e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (-3.05e7 - 2.21e7i)T + (3.77e15 + 1.16e16i)T^{2} \) |
| 31 | \( 1 + (5.00e7 - 1.54e8i)T + (-2.05e16 - 1.49e16i)T^{2} \) |
| 37 | \( 1 + (1.36e8 + 9.95e7i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (6.10e8 - 4.43e8i)T + (1.70e17 - 5.23e17i)T^{2} \) |
| 43 | \( 1 + 1.16e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (5.46e8 - 3.96e8i)T + (7.63e17 - 2.35e18i)T^{2} \) |
| 53 | \( 1 + (9.63e8 - 2.96e9i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (-4.78e9 - 3.47e9i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (-6.55e8 - 2.01e9i)T + (-3.52e19 + 2.55e19i)T^{2} \) |
| 67 | \( 1 - 1.14e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + (3.54e9 + 1.09e10i)T + (-1.86e20 + 1.35e20i)T^{2} \) |
| 73 | \( 1 + (-1.00e10 - 7.29e9i)T + (9.69e19 + 2.98e20i)T^{2} \) |
| 79 | \( 1 + (2.56e9 - 7.90e9i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (1.54e10 + 4.74e10i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 + 1.85e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-3.27e10 + 1.00e11i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
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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
−14.97336898746694693713490487974, −14.11570924281979093654042727628, −12.83917064720272020433809658476, −11.41054590259517692763124978895, −10.27952769636924292030893598699, −8.716883209533973712775085082696, −6.91334706359796733767231302242, −4.96226832403189890582725702878, −3.03257405931656070835332035876, −1.62068940816815527893074420967,
0.844485778366030246870327793111, 3.48029732157289989001976788353, 5.14107638411473838699975981227, 6.68926781200576371118319224854, 8.495396729091903804524684405083, 9.445291884061261498734038131900, 11.54029770137225176841403177664, 12.97709379450983544250781221943, 14.21828253528151948704367481653, 15.23269615890038095773995155267