| L(s) = 1 | + (6.32 + 8.70i)2-s + (−15.9 + 49.1i)4-s + (92.2 + 67.0i)5-s + (−121. − 39.3i)7-s + (125. − 40.8i)8-s + 1.22e3i·10-s + (−854. + 1.02e3i)11-s + (1.91e3 + 2.63e3i)13-s + (−423. − 1.30e3i)14-s + (3.82e3 + 2.78e3i)16-s + (−682. + 938. i)17-s + (−7.04e3 + 2.28e3i)19-s + (−4.76e3 + 3.46e3i)20-s + (−1.42e4 − 981. i)22-s + 7.16e3·23-s + ⋯ |
| L(s) = 1 | + (0.790 + 1.08i)2-s + (−0.249 + 0.768i)4-s + (0.738 + 0.536i)5-s + (−0.353 − 0.114i)7-s + (0.245 − 0.0798i)8-s + 1.22i·10-s + (−0.641 + 0.766i)11-s + (0.870 + 1.19i)13-s + (−0.154 − 0.475i)14-s + (0.934 + 0.679i)16-s + (−0.138 + 0.191i)17-s + (−1.02 + 0.333i)19-s + (−0.596 + 0.433i)20-s + (−1.34 − 0.0921i)22-s + 0.588·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.954245 + 2.94158i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.954245 + 2.94158i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (854. - 1.02e3i)T \) |
| good | 2 | \( 1 + (-6.32 - 8.70i)T + (-19.7 + 60.8i)T^{2} \) |
| 5 | \( 1 + (-92.2 - 67.0i)T + (4.82e3 + 1.48e4i)T^{2} \) |
| 7 | \( 1 + (121. + 39.3i)T + (9.51e4 + 6.91e4i)T^{2} \) |
| 13 | \( 1 + (-1.91e3 - 2.63e3i)T + (-1.49e6 + 4.59e6i)T^{2} \) |
| 17 | \( 1 + (682. - 938. i)T + (-7.45e6 - 2.29e7i)T^{2} \) |
| 19 | \( 1 + (7.04e3 - 2.28e3i)T + (3.80e7 - 2.76e7i)T^{2} \) |
| 23 | \( 1 - 7.16e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-1.49e4 - 4.84e3i)T + (4.81e8 + 3.49e8i)T^{2} \) |
| 31 | \( 1 + (3.75e4 - 2.72e4i)T + (2.74e8 - 8.44e8i)T^{2} \) |
| 37 | \( 1 + (3.12e3 - 9.60e3i)T + (-2.07e9 - 1.50e9i)T^{2} \) |
| 41 | \( 1 + (-1.45e4 + 4.74e3i)T + (3.84e9 - 2.79e9i)T^{2} \) |
| 43 | \( 1 + 1.21e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (2.67e4 + 8.22e4i)T + (-8.72e9 + 6.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.85e5 + 1.34e5i)T + (6.84e9 - 2.10e10i)T^{2} \) |
| 59 | \( 1 + (5.41e4 - 1.66e5i)T + (-3.41e10 - 2.47e10i)T^{2} \) |
| 61 | \( 1 + (-1.19e5 + 1.64e5i)T + (-1.59e10 - 4.89e10i)T^{2} \) |
| 67 | \( 1 - 5.33e4T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-5.64e5 - 4.09e5i)T + (3.95e10 + 1.21e11i)T^{2} \) |
| 73 | \( 1 + (-6.78e5 - 2.20e5i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (1.18e5 + 1.62e5i)T + (-7.51e10 + 2.31e11i)T^{2} \) |
| 83 | \( 1 + (-7.29e3 + 1.00e4i)T + (-1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 - 1.22e6T + 4.96e11T^{2} \) |
| 97 | \( 1 + (2.18e5 - 1.58e5i)T + (2.57e11 - 7.92e11i)T^{2} \) |
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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
−13.42242626322500903428447075146, −12.59418402403649634265394985946, −10.88711276733951918475049394558, −9.968781397589062560617216421825, −8.506548595821719137011737005425, −6.95953148148518964252629193578, −6.44216868062709890617317373914, −5.19289066070243444050914886583, −3.87833386551717302161731943574, −1.95095781501094605049530240943,
0.827963218979491035103161113936, 2.37116468671634032429618636813, 3.53486011587676094580736090662, 5.06803940969941257927894913502, 6.01988799440726870354424208057, 7.995409771519853094049686557081, 9.258824135441832566523540148757, 10.55551791886225137403853299446, 11.16134310812872266030426472217, 12.67080671752699612417068805976
