L(s) = 1 | + (31.5 + 5.58i)2-s + 275. i·3-s + (961. + 351. i)4-s + (−1.53e3 + 8.68e3i)6-s − 1.93e4i·7-s + (2.83e4 + 1.64e4i)8-s − 1.69e4·9-s + 1.71e5i·11-s + (−9.70e4 + 2.65e5i)12-s − 2.03e5·13-s + (1.07e5 − 6.08e5i)14-s + (8.00e5 + 6.76e5i)16-s − 2.12e6·17-s + (−5.33e5 − 9.45e4i)18-s + 5.59e5i·19-s + ⋯ |
L(s) = 1 | + (0.984 + 0.174i)2-s + 1.13i·3-s + (0.939 + 0.343i)4-s + (−0.197 + 1.11i)6-s − 1.14i·7-s + (0.864 + 0.502i)8-s − 0.286·9-s + 1.06i·11-s + (−0.389 + 1.06i)12-s − 0.547·13-s + (0.200 − 1.13i)14-s + (0.763 + 0.645i)16-s − 1.49·17-s + (−0.282 − 0.0500i)18-s + 0.225i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(0.541934 + 3.05733i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.541934 + 3.05733i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-31.5 - 5.58i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 275. iT - 5.90e4T^{2} \) |
| 7 | \( 1 + 1.93e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 1.71e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 2.03e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 2.12e6T + 2.01e12T^{2} \) |
| 19 | \( 1 - 5.59e5iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 8.97e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 6.60e6T + 4.20e14T^{2} \) |
| 31 | \( 1 - 4.19e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + 3.19e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 1.21e8T + 1.34e16T^{2} \) |
| 43 | \( 1 - 1.35e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 + 3.08e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 7.27e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + 2.51e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 9.40e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 2.07e9iT - 1.82e18T^{2} \) |
| 71 | \( 1 - 1.90e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 1.72e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 2.36e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 5.22e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 8.10e9T + 3.11e19T^{2} \) |
| 97 | \( 1 - 5.70e9T + 7.37e19T^{2} \) |
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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
−12.48629450831648406776497489840, −11.15681301093511538572877304891, −10.41195630885598500267172450216, −9.400408632594433213488726966932, −7.58900156247357046886223357509, −6.73673124143035088257387173332, −5.02045562378178888594327814927, −4.37502984030029991927896668145, −3.43160139409064096910033214779, −1.76861502029858739907536005493,
0.48433095996616068503541650723, 2.02336042608752619505830749707, 2.71703247389502986655538762842, 4.47360378108160995187077217323, 5.88295036925802328738892398301, 6.56900832608531096426232236653, 7.82137679121792477190418843618, 9.088105859731771764091399916805, 10.82625372310815903775669806070, 11.74625798174059690136639006181