L(s) = 1 | + (−5.02 − 8.70i)2-s + (−17.3 + 30.0i)3-s + (13.4 − 23.3i)4-s + (14.1 − 24.5i)5-s + 348.·6-s + (−329. + 571. i)7-s − 1.55e3·8-s + (492. + 853. i)9-s − 285.·10-s + 6.67e3·11-s + (466. + 807. i)12-s + (3.45e3 − 5.98e3i)13-s + 6.63e3·14-s + (491. + 851. i)15-s + (6.10e3 + 1.05e4i)16-s + (−7.84e3 − 1.35e4i)17-s + ⋯ |
L(s) = 1 | + (−0.444 − 0.769i)2-s + (−0.370 + 0.641i)3-s + (0.105 − 0.182i)4-s + (0.0507 − 0.0878i)5-s + 0.658·6-s + (−0.363 + 0.629i)7-s − 1.07·8-s + (0.225 + 0.390i)9-s − 0.0901·10-s + 1.51·11-s + (0.0779 + 0.134i)12-s + (0.435 − 0.755i)13-s + 0.646·14-s + (0.0376 + 0.0651i)15-s + (0.372 + 0.645i)16-s + (−0.387 − 0.670i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.25428 - 0.488476i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25428 - 0.488476i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-2.77e5 + 1.34e5i)T \) |
good | 2 | \( 1 + (5.02 + 8.70i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (17.3 - 30.0i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-14.1 + 24.5i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (329. - 571. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 - 6.67e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-3.45e3 + 5.98e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (7.84e3 + 1.35e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.32e4 + 4.02e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 - 6.93e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.17e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.24e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + (1.78e5 - 3.08e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + 9.10e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.53e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + (2.15e5 + 3.72e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-6.74e5 - 1.16e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (3.41e5 - 5.91e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.45e6 + 2.51e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.39e6 - 2.42e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 6.51e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-2.48e6 + 4.30e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (3.11e6 + 5.40e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (8.09e5 + 1.40e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.78e5T + 8.07e13T^{2} \) |
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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
−15.03612478256878156195138345933, −13.31067063054061762937222019370, −11.77718609509650929550248149153, −11.05645644713847485903709150324, −9.741445155647025969109511768085, −8.961585010869585719522049103971, −6.59167799552746026815672462518, −5.04240528886671212675697028574, −2.97197017861083339195013086412, −1.00314315102779774818700958971,
1.10350780849201996810378198650, 3.74027830421757961840042148119, 6.46455929440540191784542338427, 6.75941285135080160797748812050, 8.375615555110678950866062493757, 9.711196198287921811340214733409, 11.59664670062854996233924597516, 12.43996701540972111240650296733, 13.91359468965024969244717795374, 15.15766668871333366481172460598