| L(s) = 1 | + (−1.16 − 2.38i)2-s + (−3.09 + 3.95i)4-s + (0.0429 + 0.613i)5-s + (−1.71 + 0.0598i)7-s + (7.82 + 1.66i)8-s + (1.41 − 0.814i)10-s + (2.23 − 2.44i)11-s + (−0.903 + 0.935i)13-s + (2.13 + 4.01i)14-s + (−2.70 − 10.8i)16-s + (0.0482 − 0.459i)17-s + (0.141 − 0.664i)19-s + (−2.56 − 1.72i)20-s + (−8.42 − 2.48i)22-s + (1.57 − 4.33i)23-s + ⋯ |
| L(s) = 1 | + (−0.821 − 1.68i)2-s + (−1.54 + 1.97i)4-s + (0.0191 + 0.274i)5-s + (−0.647 + 0.0226i)7-s + (2.76 + 0.588i)8-s + (0.446 − 0.257i)10-s + (0.674 − 0.737i)11-s + (−0.250 + 0.259i)13-s + (0.570 + 1.07i)14-s + (−0.675 − 2.70i)16-s + (0.0117 − 0.111i)17-s + (0.0324 − 0.152i)19-s + (−0.572 − 0.386i)20-s + (−1.79 − 0.530i)22-s + (0.329 − 0.903i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.147804 + 0.343808i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.147804 + 0.343808i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-2.23 + 2.44i)T \) |
| good | 2 | \( 1 + (1.16 + 2.38i)T + (-1.23 + 1.57i)T^{2} \) |
| 5 | \( 1 + (-0.0429 - 0.613i)T + (-4.95 + 0.695i)T^{2} \) |
| 7 | \( 1 + (1.71 - 0.0598i)T + (6.98 - 0.488i)T^{2} \) |
| 13 | \( 1 + (0.903 - 0.935i)T + (-0.453 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.0482 + 0.459i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.141 + 0.664i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-1.57 + 4.33i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (9.02 + 4.79i)T + (16.2 + 24.0i)T^{2} \) |
| 31 | \( 1 + (4.59 - 1.31i)T + (26.2 - 16.4i)T^{2} \) |
| 37 | \( 1 + (7.04 - 1.49i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (-8.34 + 4.43i)T + (22.9 - 33.9i)T^{2} \) |
| 43 | \( 1 + (3.76 - 4.48i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (6.13 - 4.79i)T + (11.3 - 45.6i)T^{2} \) |
| 53 | \( 1 + (3.92 + 5.40i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (11.0 - 4.44i)T + (42.4 - 40.9i)T^{2} \) |
| 61 | \( 1 + (-1.57 + 5.48i)T + (-51.7 - 32.3i)T^{2} \) |
| 67 | \( 1 + (-0.459 - 2.60i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (11.1 + 1.17i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-11.9 + 10.7i)T + (7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (1.05 - 0.516i)T + (48.6 - 62.2i)T^{2} \) |
| 83 | \( 1 + (-11.5 + 11.1i)T + (2.89 - 82.9i)T^{2} \) |
| 89 | \( 1 + (11.2 + 6.51i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.82 - 0.547i)T + (96.0 + 13.4i)T^{2} \) |
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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
−9.521330114514971470578797231782, −9.197420631496239393642800667052, −8.323089157909436886952908998005, −7.29538753974838371087081456526, −6.22946614133560764197649378081, −4.70433723880478481580776913963, −3.60635222795490998637759064111, −2.94896192512930289695232533292, −1.73723301621650206621858882610, −0.24893316517377555518543929216,
1.48072926754251656816436224944, 3.66640116526607888922815781798, 4.94165368630765279648190845577, 5.62497256482849745199538711787, 6.64712131973367646483278510476, 7.20598025089536017939916979887, 7.957381037213862804810267606896, 9.177134718284853383881746652672, 9.255185940355130894813123978836, 10.19098210422516848943554195350
