L(s) = 1 | + (172. − 54.1i)2-s + (−3.10e3 − 2.16e3i)3-s + (2.69e4 − 1.86e4i)4-s − 1.92e5i·5-s + (−6.53e5 − 2.06e5i)6-s + 3.66e5i·7-s + (3.63e6 − 4.68e6i)8-s + (4.93e6 + 1.34e7i)9-s + (−1.04e7 − 3.32e7i)10-s − 5.82e7·11-s + (−1.24e8 − 3.44e5i)12-s − 2.65e8·13-s + (1.98e7 + 6.33e7i)14-s + (−4.17e8 + 5.97e8i)15-s + (3.75e8 − 1.00e9i)16-s − 1.79e9i·17-s + ⋯ |
L(s) = 1 | + (0.954 − 0.298i)2-s + (−0.819 − 0.572i)3-s + (0.821 − 0.570i)4-s − 1.10i·5-s + (−0.953 − 0.301i)6-s + 0.168i·7-s + (0.613 − 0.789i)8-s + (0.344 + 0.938i)9-s + (−0.328 − 1.05i)10-s − 0.901·11-s + (−0.999 − 0.00277i)12-s − 1.17·13-s + (0.0503 + 0.160i)14-s + (−0.630 + 0.902i)15-s + (0.349 − 0.937i)16-s − 1.05i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00277i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(8)\) |
\(\approx\) |
\(0.00223875 + 1.61505i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00223875 + 1.61505i\) |
\(L(\frac{17}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-172. + 54.1i)T \) |
| 3 | \( 1 + (3.10e3 + 2.16e3i)T \) |
good | 5 | \( 1 + 1.92e5iT - 3.05e10T^{2} \) |
| 7 | \( 1 - 3.66e5iT - 4.74e12T^{2} \) |
| 11 | \( 1 + 5.82e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 2.65e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.79e9iT - 2.86e18T^{2} \) |
| 19 | \( 1 - 4.75e9iT - 1.51e19T^{2} \) |
| 23 | \( 1 + 1.92e10T + 2.66e20T^{2} \) |
| 29 | \( 1 + 1.16e11iT - 8.62e21T^{2} \) |
| 31 | \( 1 - 1.69e11iT - 2.34e22T^{2} \) |
| 37 | \( 1 - 7.59e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 1.15e12iT - 1.55e24T^{2} \) |
| 43 | \( 1 + 1.91e12iT - 3.17e24T^{2} \) |
| 47 | \( 1 - 1.35e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 7.61e12iT - 7.31e25T^{2} \) |
| 59 | \( 1 + 1.15e12T + 3.65e26T^{2} \) |
| 61 | \( 1 + 2.88e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 7.60e13iT - 2.46e27T^{2} \) |
| 71 | \( 1 - 6.50e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.24e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + 5.22e13iT - 2.91e28T^{2} \) |
| 83 | \( 1 + 3.24e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 1.32e14iT - 1.74e29T^{2} \) |
| 97 | \( 1 - 8.29e14T + 6.33e29T^{2} \) |
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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
−15.81679280220146895720450747792, −13.86805149781720999030845497855, −12.58027691023157139189754504000, −11.91975100274333700331655546861, −10.14480598921872191003402596154, −7.59337029754687050116075664306, −5.70864996654724345579182696847, −4.70051839647052647808933536008, −2.14080363128269988452175607785, −0.46910028136008476455112447387,
2.76278762495772741524920208719, 4.47492141402700236901758233935, 6.00617432356741649159895918420, 7.35543922122615962883413723932, 10.28606835270669301546363534100, 11.32739824486160191964530549666, 12.81931251139604533979477134279, 14.59206264226792740031230290161, 15.45387755596870453467313936349, 16.83009148562485667596460758418