| L(s) = 1 | + (−0.929 − 2.65i)2-s + (−3.06 + 2.44i)4-s + (−1.35 + 2.82i)5-s + (2.26 − 2.83i)7-s + (−0.186 − 0.117i)8-s + (8.75 + 0.986i)10-s + (−9.83 + 6.17i)11-s + (−18.5 − 4.23i)13-s + (−9.64 − 3.37i)14-s + (−3.62 + 15.8i)16-s + (12.3 + 12.3i)17-s + (−2.90 + 25.7i)19-s + (−2.73 − 11.9i)20-s + (25.5 + 20.3i)22-s + (−9.50 + 4.57i)23-s + ⋯ |
| L(s) = 1 | + (−0.464 − 1.32i)2-s + (−0.766 + 0.611i)4-s + (−0.271 + 0.564i)5-s + (0.323 − 0.405i)7-s + (−0.0232 − 0.0146i)8-s + (0.875 + 0.0986i)10-s + (−0.893 + 0.561i)11-s + (−1.42 − 0.325i)13-s + (−0.688 − 0.240i)14-s + (−0.226 + 0.993i)16-s + (0.725 + 0.725i)17-s + (−0.152 + 1.35i)19-s + (−0.136 − 0.598i)20-s + (1.16 + 0.926i)22-s + (−0.413 + 0.198i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.240396 + 0.156410i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.240396 + 0.156410i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + (14.7 + 24.9i)T \) |
| good | 2 | \( 1 + (0.929 + 2.65i)T + (-3.12 + 2.49i)T^{2} \) |
| 5 | \( 1 + (1.35 - 2.82i)T + (-15.5 - 19.5i)T^{2} \) |
| 7 | \( 1 + (-2.26 + 2.83i)T + (-10.9 - 47.7i)T^{2} \) |
| 11 | \( 1 + (9.83 - 6.17i)T + (52.4 - 109. i)T^{2} \) |
| 13 | \( 1 + (18.5 + 4.23i)T + (152. + 73.3i)T^{2} \) |
| 17 | \( 1 + (-12.3 - 12.3i)T + 289iT^{2} \) |
| 19 | \( 1 + (2.90 - 25.7i)T + (-351. - 80.3i)T^{2} \) |
| 23 | \( 1 + (9.50 - 4.57i)T + (329. - 413. i)T^{2} \) |
| 31 | \( 1 + (8.22 + 23.5i)T + (-751. + 599. i)T^{2} \) |
| 37 | \( 1 + (25.3 + 15.9i)T + (593. + 1.23e3i)T^{2} \) |
| 41 | \( 1 + (36.7 - 36.7i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-37.8 - 13.2i)T + (1.44e3 + 1.15e3i)T^{2} \) |
| 47 | \( 1 + (16.7 + 26.6i)T + (-958. + 1.99e3i)T^{2} \) |
| 53 | \( 1 + (-47.4 - 22.8i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 + 2.51T + 3.48e3T^{2} \) |
| 61 | \( 1 + (29.2 - 3.29i)T + (3.62e3 - 828. i)T^{2} \) |
| 67 | \( 1 + (127. - 29.0i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (73.2 + 16.7i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-8.81 + 25.2i)T + (-4.16e3 - 3.32e3i)T^{2} \) |
| 79 | \( 1 + (-51.2 + 81.5i)T + (-2.70e3 - 5.62e3i)T^{2} \) |
| 83 | \( 1 + (1.49 + 1.87i)T + (-1.53e3 + 6.71e3i)T^{2} \) |
| 89 | \( 1 + (-28.8 - 82.3i)T + (-6.19e3 + 4.93e3i)T^{2} \) |
| 97 | \( 1 + (57.2 + 6.45i)T + (9.17e3 + 2.09e3i)T^{2} \) |
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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
−11.92240859030753724568030869280, −10.74732479229704638427079443107, −10.26414381934771937244169529571, −9.556794400479040659356901154446, −8.034404237823693166819516613178, −7.38471096034761866273323127031, −5.74396933328010453450118989398, −4.21171923931647232162082888826, −3.01423772754145348737628171248, −1.81154507365738388411937312876,
0.16207533209569646781444653242, 2.71717879324192224517812890324, 4.91567181346562242086981816914, 5.37955134943330830657131813873, 6.87538002768938257480301517638, 7.59410187471965134864905305118, 8.575840516265755402127760912960, 9.182594118622543299060479615492, 10.40894773955759422157505118127, 11.75195258343982791590103358285
