| L(s) = 1 | + (51.7 − 37.6i)2-s + (−252. − 777. i)3-s + (1.26e3 − 3.89e3i)4-s + (2.51e4 + 1.82e4i)5-s + (−4.23e4 − 3.07e4i)6-s + (2.92e4 − 8.99e4i)7-s + (−8.10e4 − 2.49e5i)8-s + (7.48e5 − 5.44e5i)9-s + 1.98e6·10-s + (5.31e6 + 2.50e6i)11-s − 3.34e6·12-s + (6.79e6 − 4.93e6i)13-s + (−1.87e6 − 5.75e6i)14-s + (7.85e6 − 2.41e7i)15-s + (−1.35e7 − 9.86e6i)16-s + (−4.48e7 − 3.25e7i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.200 − 0.615i)3-s + (0.154 − 0.475i)4-s + (0.719 + 0.522i)5-s + (−0.370 − 0.269i)6-s + (0.0938 − 0.288i)7-s + (−0.109 − 0.336i)8-s + (0.469 − 0.341i)9-s + 0.628·10-s + (0.904 + 0.426i)11-s − 0.323·12-s + (0.390 − 0.283i)13-s + (−0.0663 − 0.204i)14-s + (0.177 − 0.547i)15-s + (−0.202 − 0.146i)16-s + (−0.450 − 0.327i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(3.023776586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.023776586\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-51.7 + 37.6i)T \) |
| 11 | \( 1 + (-5.31e6 - 2.50e6i)T \) |
| good | 3 | \( 1 + (252. + 777. i)T + (-1.28e6 + 9.37e5i)T^{2} \) |
| 5 | \( 1 + (-2.51e4 - 1.82e4i)T + (3.77e8 + 1.16e9i)T^{2} \) |
| 7 | \( 1 + (-2.92e4 + 8.99e4i)T + (-7.83e10 - 5.69e10i)T^{2} \) |
| 13 | \( 1 + (-6.79e6 + 4.93e6i)T + (9.35e13 - 2.88e14i)T^{2} \) |
| 17 | \( 1 + (4.48e7 + 3.25e7i)T + (3.06e15 + 9.41e15i)T^{2} \) |
| 19 | \( 1 + (5.25e7 + 1.61e8i)T + (-3.40e16 + 2.47e16i)T^{2} \) |
| 23 | \( 1 + 8.42e7T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-4.54e7 + 1.39e8i)T + (-8.30e18 - 6.03e18i)T^{2} \) |
| 31 | \( 1 + (-1.82e9 + 1.32e9i)T + (7.54e18 - 2.32e19i)T^{2} \) |
| 37 | \( 1 + (-5.46e9 + 1.68e10i)T + (-1.97e20 - 1.43e20i)T^{2} \) |
| 41 | \( 1 + (9.26e9 + 2.85e10i)T + (-7.48e20 + 5.43e20i)T^{2} \) |
| 43 | \( 1 + 2.30e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-3.16e10 - 9.73e10i)T + (-4.41e21 + 3.20e21i)T^{2} \) |
| 53 | \( 1 + (6.79e10 - 4.93e10i)T + (8.04e21 - 2.47e22i)T^{2} \) |
| 59 | \( 1 + (1.71e11 - 5.26e11i)T + (-8.49e22 - 6.17e22i)T^{2} \) |
| 61 | \( 1 + (-3.82e11 - 2.77e11i)T + (5.00e22 + 1.53e23i)T^{2} \) |
| 67 | \( 1 - 1.20e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + (3.76e11 + 2.73e11i)T + (3.60e23 + 1.10e24i)T^{2} \) |
| 73 | \( 1 + (-4.50e11 + 1.38e12i)T + (-1.35e24 - 9.82e23i)T^{2} \) |
| 79 | \( 1 + (1.13e12 - 8.27e11i)T + (1.44e24 - 4.43e24i)T^{2} \) |
| 83 | \( 1 + (-1.22e12 - 8.87e11i)T + (2.74e24 + 8.43e24i)T^{2} \) |
| 89 | \( 1 + 9.25e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + (4.51e12 - 3.28e12i)T + (2.07e25 - 6.40e25i)T^{2} \) |
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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
−14.22649806597737343499584528754, −13.25888058095409617620860955445, −12.08830439189323001488963081228, −10.74990567577505643237793784495, −9.402757306149548582974249177659, −7.09039806019535384035820878705, −6.08983160236495875720133454270, −4.16560885864617823903237728323, −2.30821086335739625310346082909, −0.955172489157033769965710680782,
1.67238722593671556517906096468, 3.88045896882108856101508871459, 5.18826531148128266209081408366, 6.45722338354557304660048840569, 8.482581015195399450880601381418, 9.837813818086362130977794991490, 11.41900230961304273425918112616, 12.89654412558151920039925615851, 13.98671600002401605749483485378, 15.29978085530000635019433262498
