| L(s) = 1 | + (−3.99 − 2.90i)2-s + (−6.95 − 9.56i)3-s + (2.58 + 7.95i)4-s − 0.0618·5-s + 58.3i·6-s + (5.65 + 17.3i)7-s + (−11.6 + 35.8i)8-s + (−18.1 + 55.9i)9-s + (0.247 + 0.179i)10-s + (−53.0 + 17.2i)11-s + (58.1 − 79.9i)12-s + (−35.4 − 48.7i)13-s + (27.8 − 85.8i)14-s + (0.430 + 0.592i)15-s + (258. − 188. i)16-s + (−449. − 145. i)17-s + ⋯ |
| L(s) = 1 | + (−0.998 − 0.725i)2-s + (−0.772 − 1.06i)3-s + (0.161 + 0.496i)4-s − 0.00247·5-s + 1.62i·6-s + (0.115 + 0.355i)7-s + (−0.182 + 0.560i)8-s + (−0.224 + 0.691i)9-s + (0.00247 + 0.00179i)10-s + (−0.438 + 0.142i)11-s + (0.403 − 0.555i)12-s + (−0.209 − 0.288i)13-s + (0.142 − 0.438i)14-s + (0.00191 + 0.00263i)15-s + (1.01 − 0.734i)16-s + (−1.55 − 0.504i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0943269 + 0.143969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0943269 + 0.143969i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-653. - 704. i)T \) |
| good | 2 | \( 1 + (3.99 + 2.90i)T + (4.94 + 15.2i)T^{2} \) |
| 3 | \( 1 + (6.95 + 9.56i)T + (-25.0 + 77.0i)T^{2} \) |
| 5 | \( 1 + 0.0618T + 625T^{2} \) |
| 7 | \( 1 + (-5.65 - 17.3i)T + (-1.94e3 + 1.41e3i)T^{2} \) |
| 11 | \( 1 + (53.0 - 17.2i)T + (1.18e4 - 8.60e3i)T^{2} \) |
| 13 | \( 1 + (35.4 + 48.7i)T + (-8.82e3 + 2.71e4i)T^{2} \) |
| 17 | \( 1 + (449. + 145. i)T + (6.75e4 + 4.90e4i)T^{2} \) |
| 19 | \( 1 + (-209. - 151. i)T + (4.02e4 + 1.23e5i)T^{2} \) |
| 23 | \( 1 + (340. + 110. i)T + (2.26e5 + 1.64e5i)T^{2} \) |
| 29 | \( 1 + (301. - 415. i)T + (-2.18e5 - 6.72e5i)T^{2} \) |
| 37 | \( 1 + 1.30e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (2.22e3 + 1.61e3i)T + (8.73e5 + 2.68e6i)T^{2} \) |
| 43 | \( 1 + (-641. + 883. i)T + (-1.05e6 - 3.25e6i)T^{2} \) |
| 47 | \( 1 + (-3.22e3 + 2.34e3i)T + (1.50e6 - 4.64e6i)T^{2} \) |
| 53 | \( 1 + (2.02e3 + 659. i)T + (6.38e6 + 4.63e6i)T^{2} \) |
| 59 | \( 1 + (1.78e3 - 1.29e3i)T + (3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 - 1.49e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 1.07e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-501. + 1.54e3i)T + (-2.05e7 - 1.49e7i)T^{2} \) |
| 73 | \( 1 + (3.90e3 - 1.26e3i)T + (2.29e7 - 1.66e7i)T^{2} \) |
| 79 | \( 1 + (6.34e3 + 2.06e3i)T + (3.15e7 + 2.28e7i)T^{2} \) |
| 83 | \( 1 + (5.67e3 - 7.80e3i)T + (-1.46e7 - 4.51e7i)T^{2} \) |
| 89 | \( 1 + (-1.25e4 + 4.08e3i)T + (5.07e7 - 3.68e7i)T^{2} \) |
| 97 | \( 1 + (5.20e3 + 1.60e4i)T + (-7.16e7 + 5.20e7i)T^{2} \) |
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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
−15.56384978727947010046712037111, −13.76280813275230404983006662346, −12.33491991425532079509895318152, −11.53829835302064714084198025394, −10.31384568051278140421248264066, −8.793402337209926487033252053699, −7.31528472424556730923351396567, −5.60154274891642666320761743972, −2.03830703942407319852333487442, −0.18516042088579383672824087117,
4.34762087003864100172116358749, 6.17919090400483482250450350696, 7.81487016845882102604974953575, 9.335466561963120786787333867852, 10.33293790440677784456758713280, 11.53597124002506914785934885525, 13.42551357152947704275316864500, 15.39735807303077028672336317037, 15.86522569039532379533481377082, 17.07382524129513590298034227359
