| L(s) = 1 | + (8.47 + 3.08i)2-s + (8.38 + 13.1i)3-s + (37.7 + 31.6i)4-s + (17.5 − 99.7i)5-s + (30.5 + 137. i)6-s + (−143. + 120. i)7-s + (77.8 + 134. i)8-s + (−102. + 220. i)9-s + (456. − 790. i)10-s + (−51.6 − 292. i)11-s + (−99.5 + 761. i)12-s + (169. − 61.5i)13-s + (−1.59e3 + 579. i)14-s + (1.45e3 − 605. i)15-s + (−29.9 − 169. i)16-s + (81.3 − 140. i)17-s + ⋯ |
| L(s) = 1 | + (1.49 + 0.545i)2-s + (0.538 + 0.842i)3-s + (1.17 + 0.989i)4-s + (0.314 − 1.78i)5-s + (0.346 + 1.55i)6-s + (−1.11 + 0.931i)7-s + (0.430 + 0.745i)8-s + (−0.420 + 0.907i)9-s + (1.44 − 2.50i)10-s + (−0.128 − 0.729i)11-s + (−0.199 + 1.52i)12-s + (0.277 − 0.100i)13-s + (−2.17 + 0.789i)14-s + (1.67 − 0.694i)15-s + (−0.0292 − 0.165i)16-s + (0.0682 − 0.118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.01642 + 1.43840i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.01642 + 1.43840i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-8.38 - 13.1i)T \) |
| good | 2 | \( 1 + (-8.47 - 3.08i)T + (24.5 + 20.5i)T^{2} \) |
| 5 | \( 1 + (-17.5 + 99.7i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (143. - 120. i)T + (2.91e3 - 1.65e4i)T^{2} \) |
| 11 | \( 1 + (51.6 + 292. i)T + (-1.51e5 + 5.50e4i)T^{2} \) |
| 13 | \( 1 + (-169. + 61.5i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-81.3 + 140. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-441. - 764. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.85e3 - 1.55e3i)T + (1.11e6 + 6.33e6i)T^{2} \) |
| 29 | \( 1 + (-317. - 115. i)T + (1.57e7 + 1.31e7i)T^{2} \) |
| 31 | \( 1 + (-4.27e3 - 3.58e3i)T + (4.97e6 + 2.81e7i)T^{2} \) |
| 37 | \( 1 + (4.30e3 - 7.46e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (2.21e3 - 806. i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 + (1.06e3 + 6.06e3i)T + (-1.38e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (5.58e3 - 4.68e3i)T + (3.98e7 - 2.25e8i)T^{2} \) |
| 53 | \( 1 + 4.76e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (3.61e3 - 2.04e4i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (-7.02e3 + 5.89e3i)T + (1.46e8 - 8.31e8i)T^{2} \) |
| 67 | \( 1 + (-3.54e4 + 1.29e4i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (-1.14e4 + 1.98e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (1.41e4 + 2.44e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (6.48e4 + 2.36e4i)T + (2.35e9 + 1.97e9i)T^{2} \) |
| 83 | \( 1 + (-1.12e5 - 4.08e4i)T + (3.01e9 + 2.53e9i)T^{2} \) |
| 89 | \( 1 + (5.25e4 + 9.10e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-7.02e3 - 3.98e4i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
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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
−16.03215682290074614463801726390, −15.44166603021501922864341295442, −13.81256742981671048655800299641, −13.06603447779415527487828727807, −12.02364276099593260901607260421, −9.575008611334663037688188155107, −8.506441817244838028240921419399, −5.86176566550131042961624287876, −4.93595023409701837208885993609, −3.29216712619914632832291613604,
2.53139655389627142424472378097, 3.59162271771418632333841839371, 6.36642113709559832499305826630, 7.09978640770948165435512012072, 10.02232096223014351516110002843, 11.25304929121152978759530461070, 12.76226547103090963488267709443, 13.62280274027581240180532846910, 14.39595802855830096504710913941, 15.36270865319652032479832207876
