L(s) = 1 | + (83.3 + 160. i)2-s + (−3.55e3 − 1.30e3i)3-s + (−1.88e4 + 2.67e4i)4-s − 9.33e4i·5-s + (−8.62e4 − 6.80e5i)6-s − 1.14e6i·7-s + (−5.87e6 − 8.04e5i)8-s + (1.09e7 + 9.29e6i)9-s + (1.50e7 − 7.77e6i)10-s + 1.80e6·11-s + (1.02e8 − 7.05e7i)12-s + 3.17e8·13-s + (1.84e8 − 9.54e7i)14-s + (−1.21e8 + 3.31e8i)15-s + (−3.60e8 − 1.01e9i)16-s + 2.67e9i·17-s + ⋯ |
L(s) = 1 | + (0.460 + 0.887i)2-s + (−0.938 − 0.344i)3-s + (−0.576 + 0.817i)4-s − 0.534i·5-s + (−0.125 − 0.992i)6-s − 0.525i·7-s + (−0.990 − 0.135i)8-s + (0.761 + 0.647i)9-s + (0.474 − 0.245i)10-s + 0.0279·11-s + (0.822 − 0.568i)12-s + 1.40·13-s + (0.466 − 0.241i)14-s + (−0.184 + 0.501i)15-s + (−0.335 − 0.942i)16-s + 1.58i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(8)\) |
\(\approx\) |
\(1.56865 + 0.488944i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56865 + 0.488944i\) |
\(L(\frac{17}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-83.3 - 160. i)T \) |
| 3 | \( 1 + (3.55e3 + 1.30e3i)T \) |
good | 5 | \( 1 + 9.33e4iT - 3.05e10T^{2} \) |
| 7 | \( 1 + 1.14e6iT - 4.74e12T^{2} \) |
| 11 | \( 1 - 1.80e6T + 4.17e15T^{2} \) |
| 13 | \( 1 - 3.17e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 2.67e9iT - 2.86e18T^{2} \) |
| 19 | \( 1 + 2.07e9iT - 1.51e19T^{2} \) |
| 23 | \( 1 - 2.43e10T + 2.66e20T^{2} \) |
| 29 | \( 1 + 4.77e9iT - 8.62e21T^{2} \) |
| 31 | \( 1 + 5.30e10iT - 2.34e22T^{2} \) |
| 37 | \( 1 - 4.06e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 1.36e12iT - 1.55e24T^{2} \) |
| 43 | \( 1 + 2.85e12iT - 3.17e24T^{2} \) |
| 47 | \( 1 + 7.27e11T + 1.20e25T^{2} \) |
| 53 | \( 1 - 7.05e12iT - 7.31e25T^{2} \) |
| 59 | \( 1 + 1.63e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 3.06e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 7.17e13iT - 2.46e27T^{2} \) |
| 71 | \( 1 - 4.82e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 9.56e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 3.27e14iT - 2.91e28T^{2} \) |
| 83 | \( 1 - 2.88e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 2.05e14iT - 1.74e29T^{2} \) |
| 97 | \( 1 + 5.41e14T + 6.33e29T^{2} \) |
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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
−16.66063046057353768586739607174, −15.38537491374520378398966152357, −13.48965082579827465132428967100, −12.63131341375103406775666959704, −10.91466956548332389298588688700, −8.590569804608623272596445337617, −6.93261108959974899590168874086, −5.62389314824031548796867522614, −4.10793344690011006651519333492, −0.898348320189514778052369284822,
0.997045603762658054585504930147, 3.15653429637685260959500703716, 4.93412905666406630679758658646, 6.36638212607437327563893510691, 9.296557686221597940065608331389, 10.80896673245218678524678187408, 11.65524582151033689503177700365, 13.12439618735915539656739891377, 14.78324031399392503941153598244, 16.14129407084186597165660411202