| L(s) = 1 | + (−0.167 + 0.0449i)2-s + (2.09 + 0.560i)3-s + (−13.8 + 7.98i)4-s + (−20.4 + 14.3i)5-s − 0.376·6-s + (20.9 + 44.2i)7-s + (3.92 − 3.92i)8-s + (−66.0 − 38.1i)9-s + (2.79 − 3.33i)10-s + (58.3 + 101. i)11-s + (−33.3 + 8.94i)12-s + (−115. + 115. i)13-s + (−5.51 − 6.49i)14-s + (−50.8 + 18.5i)15-s + (127. − 220. i)16-s + (87.3 − 325. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0419 + 0.0112i)2-s + (0.232 + 0.0622i)3-s + (−0.864 + 0.499i)4-s + (−0.818 + 0.573i)5-s − 0.0104·6-s + (0.428 + 0.903i)7-s + (0.0614 − 0.0614i)8-s + (−0.815 − 0.471i)9-s + (0.0279 − 0.0333i)10-s + (0.482 + 0.835i)11-s + (−0.231 + 0.0621i)12-s + (−0.681 + 0.681i)13-s + (−0.0281 − 0.0331i)14-s + (−0.225 + 0.0823i)15-s + (0.497 − 0.861i)16-s + (0.302 − 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 - 0.811i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.583 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.375780 + 0.733121i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.375780 + 0.733121i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (20.4 - 14.3i)T \) |
| 7 | \( 1 + (-20.9 - 44.2i)T \) |
| good | 2 | \( 1 + (0.167 - 0.0449i)T + (13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (-2.09 - 0.560i)T + (70.1 + 40.5i)T^{2} \) |
| 11 | \( 1 + (-58.3 - 101. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (115. - 115. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (-87.3 + 325. i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-254. - 146. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-33.6 - 125. i)T + (-2.42e5 + 1.39e5i)T^{2} \) |
| 29 | \( 1 - 1.16e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + (-489. - 848. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (1.59e3 - 426. i)T + (1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 - 1.69e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (2.40e3 - 2.40e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-1.48e3 + 398. i)T + (4.22e6 - 2.43e6i)T^{2} \) |
| 53 | \( 1 + (-798. - 213. i)T + (6.83e6 + 3.94e6i)T^{2} \) |
| 59 | \( 1 + (-5.36e3 + 3.09e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-102. + 178. i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-306. + 1.14e3i)T + (-1.74e7 - 1.00e7i)T^{2} \) |
| 71 | \( 1 + 1.52e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (4.00e3 + 1.07e3i)T + (2.45e7 + 1.41e7i)T^{2} \) |
| 79 | \( 1 + (2.65e3 + 1.53e3i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-6.15e3 + 6.15e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (1.42e3 + 824. i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-5.26e3 - 5.26e3i)T + 8.85e7iT^{2} \) |
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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
−16.15168989890363056899037897008, −14.70922234532549471492731404729, −14.21178237048203885220888212851, −12.20981723934305693697268128447, −11.72577175083742425374174477670, −9.589276976755936290505961694434, −8.562563214251969927153557771623, −7.20291143432748817135053898779, −4.90985627628197684655803074601, −3.17256877066416982154982126971,
0.60587259966898558782679962430, 3.94579359682676226677774332567, 5.43274470107896574892633829726, 7.83357592146006704074078530894, 8.710825022880647150849349200194, 10.33188076429071437856832994228, 11.61371582219842904560567033588, 13.20972282801231094096466777354, 14.11821391691739212955546065687, 15.16605530768258779854450235877
