L(s) = 1 | + (7.22 + 8.70i)2-s + (6.18 − 14.9i)3-s + (−23.4 + 125. i)4-s + (324. − 134. i)5-s + (174. − 54.1i)6-s + (441. − 441. i)7-s + (−1.26e3 + 705. i)8-s + (1.36e3 + 1.36e3i)9-s + (3.51e3 + 1.85e3i)10-s + (668. + 1.61e3i)11-s + (1.73e3 + 1.12e3i)12-s + (2.81e3 + 1.16e3i)13-s + (7.03e3 + 650. i)14-s − 5.67e3i·15-s + (−1.52e4 − 5.90e3i)16-s + 1.44e4i·17-s + ⋯ |
L(s) = 1 | + (0.638 + 0.769i)2-s + (0.132 − 0.319i)3-s + (−0.183 + 0.983i)4-s + (1.16 − 0.480i)5-s + (0.330 − 0.102i)6-s + (0.486 − 0.486i)7-s + (−0.873 + 0.487i)8-s + (0.622 + 0.622i)9-s + (1.11 + 0.585i)10-s + (0.151 + 0.365i)11-s + (0.289 + 0.188i)12-s + (0.354 + 0.146i)13-s + (0.685 + 0.0633i)14-s − 0.434i·15-s + (−0.932 − 0.360i)16-s + 0.711i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.74147 + 1.34707i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.74147 + 1.34707i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-7.22 - 8.70i)T \) |
good | 3 | \( 1 + (-6.18 + 14.9i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-324. + 134. i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-441. + 441. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (-668. - 1.61e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (-2.81e3 - 1.16e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 1.44e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-7.86e3 - 3.25e3i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (2.27e4 + 2.27e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (-8.77e4 + 2.11e5i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 1.47e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (2.26e5 - 9.37e4i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (2.08e5 + 2.08e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (3.15e5 + 7.60e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 + 1.10e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-7.88e5 - 1.90e6i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (2.03e6 - 8.43e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (5.94e5 - 1.43e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (2.14e5 - 5.16e5i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (1.56e6 - 1.56e6i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (-4.37e5 - 4.37e5i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 + 4.34e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (5.99e6 + 2.48e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (7.82e5 - 7.82e5i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 + 4.43e6T + 8.07e13T^{2} \) |
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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
−15.34765459179879181304837987457, −13.88573334101211889954894095019, −13.45125410152939905863032817284, −12.20725569652575074800356444747, −10.25490654343597095066065199574, −8.614919238962348260835871404465, −7.25746289890984771395093899301, −5.78818568939902764955084929668, −4.36735918536527778463325535171, −1.84700120845042883898472675460,
1.55960611780638816728189902666, 3.21478240921639858736192134824, 5.12432406309236670621304850389, 6.47304870577626532825234066433, 9.116609893025749748067945296858, 10.08489316116065338898584656578, 11.30283508835827111846712154550, 12.67922570619611829087321553913, 13.91880229222048530801712506421, 14.69418558200616857605655937231