L(s) = 1 | + (−6.23 + 10.8i)2-s + (−13.5 + 7.79i)3-s + (−45.8 − 79.3i)4-s + (175. + 101. i)5-s − 194. i·6-s + 345.·8-s + (121.5 − 210. i)9-s + (−2.18e3 + 1.26e3i)10-s + (−437. − 758. i)11-s + (1.23e3 + 714. i)12-s − 275. i·13-s − 3.15e3·15-s + (780. − 1.35e3i)16-s + (3.79e3 − 2.19e3i)17-s + (1.51e3 + 2.62e3i)18-s + (1.16e4 + 6.75e3i)19-s + ⋯ |
L(s) = 1 | + (−0.779 + 1.35i)2-s + (−0.5 + 0.288i)3-s + (−0.716 − 1.24i)4-s + (1.40 + 0.809i)5-s − 0.900i·6-s + 0.674·8-s + (0.166 − 0.288i)9-s + (−2.18 + 1.26i)10-s + (−0.328 − 0.569i)11-s + (0.716 + 0.413i)12-s − 0.125i·13-s − 0.935·15-s + (0.190 − 0.329i)16-s + (0.773 − 0.446i)17-s + (0.259 + 0.450i)18-s + (1.70 + 0.984i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.457450446\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.457450446\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (13.5 - 7.79i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (6.23 - 10.8i)T + (-32 - 55.4i)T^{2} \) |
| 5 | \( 1 + (-175. - 101. i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (437. + 758. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + 275. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-3.79e3 + 2.19e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-1.16e4 - 6.75e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-6.36e3 + 1.10e4i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 - 6.26e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.76e4 + 1.02e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (7.86e3 - 1.36e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + 6.99e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.13e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-4.01e4 - 2.31e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (6.40e4 + 1.10e5i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (2.42e5 - 1.40e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (8.47e4 + 4.89e4i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (8.72e4 + 1.51e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 3.45e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-1.04e5 + 6.03e4i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (3.81e5 - 6.61e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 8.59e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.55e5 - 8.99e4i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 3.40e5iT - 8.32e11T^{2} \) |
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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
−12.25026510112081264466126899433, −10.78348264755187789265609502645, −9.942487999970711415150876472285, −9.299185413106726030055072785678, −7.903613173731166387365184466726, −6.85094126762745032787062740352, −5.88795683465937307749379694461, −5.34247919943477163163211652340, −2.95810999235717902687315655922, −0.923014931527965953673955345324,
0.897265493756241526355295301837, 1.61115997845326732574353243943, 2.89248258070465242086191437765, 4.89694826542435441375125652100, 5.93501985582168910459762295712, 7.58878289349634906438300195922, 9.047033764386503498020529295029, 9.644853843749638283978502121470, 10.43348353312411349225410250500, 11.55003037969726956173501686913