| L(s) = 1 | + (−0.732 + 2.73i)2-s + (−4.74 − 8.22i)3-s + (−6.92 − 4i)4-s + (−19.2 − 19.2i)5-s + (25.9 − 6.95i)6-s + (−5.11 − 19.0i)7-s + (16 − 15.9i)8-s + (−4.58 + 7.94i)9-s + (66.6 − 38.4i)10-s + (−31.7 − 8.51i)11-s + 75.9i·12-s + (−94.3 + 140. i)13-s + 55.8·14-s + (−66.8 + 249. i)15-s + (31.9 + 55.4i)16-s + (−186. − 107. i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 + 0.683i)2-s + (−0.527 − 0.913i)3-s + (−0.433 − 0.250i)4-s + (−0.769 − 0.769i)5-s + (0.720 − 0.193i)6-s + (−0.104 − 0.389i)7-s + (0.250 − 0.249i)8-s + (−0.0566 + 0.0980i)9-s + (0.666 − 0.384i)10-s + (−0.262 − 0.0704i)11-s + 0.527i·12-s + (−0.558 + 0.829i)13-s + 0.285·14-s + (−0.297 + 1.10i)15-s + (0.124 + 0.216i)16-s + (−0.644 − 0.371i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.380795 - 0.511956i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.380795 - 0.511956i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.732 - 2.73i)T \) |
| 13 | \( 1 + (94.3 - 140. i)T \) |
| good | 3 | \( 1 + (4.74 + 8.22i)T + (-40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (19.2 + 19.2i)T + 625iT^{2} \) |
| 7 | \( 1 + (5.11 + 19.0i)T + (-2.07e3 + 1.20e3i)T^{2} \) |
| 11 | \( 1 + (31.7 + 8.51i)T + (1.26e4 + 7.32e3i)T^{2} \) |
| 17 | \( 1 + (186. + 107. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-407. + 109. i)T + (1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-833. + 481. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (772. + 1.33e3i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-750. - 750. i)T + 9.23e5iT^{2} \) |
| 37 | \( 1 + (-680. - 182. i)T + (1.62e6 + 9.37e5i)T^{2} \) |
| 41 | \( 1 + (-146. + 548. i)T + (-2.44e6 - 1.41e6i)T^{2} \) |
| 43 | \( 1 + (141. + 81.8i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (1.73e3 - 1.73e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + 4.53e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (201. + 752. i)T + (-1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (1.97e3 - 3.41e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-1.78e3 + 6.65e3i)T + (-1.74e7 - 1.00e7i)T^{2} \) |
| 71 | \( 1 + (-5.33e3 + 1.42e3i)T + (2.20e7 - 1.27e7i)T^{2} \) |
| 73 | \( 1 + (2.49e3 - 2.49e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 - 6.71e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (-1.36e3 - 1.36e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + (-6.03e3 - 1.61e3i)T + (5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (4.67e3 - 1.25e3i)T + (7.66e7 - 4.42e7i)T^{2} \) |
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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
−16.49406579630601334280714780814, −15.36258230643681257444628228609, −13.69252869103118036563249607261, −12.57079597549077315455729758681, −11.43150256968859523275556317124, −9.310413588766681391567685209486, −7.73909223021353065904058039495, −6.66507814418906343841553378425, −4.71756580313311405627036434338, −0.59049342268376283141833400199,
3.30550417893922167901907340829, 5.13260098668042408151671593155, 7.57014967056757253442461289059, 9.499526694989230731260494837712, 10.73859210779971136293303180361, 11.48804824714714505828700896385, 12.99761141444058179631494732739, 14.92170419826579683270852527364, 15.72630762598155197559649682280, 17.09969270189128609506678438254
