| L(s) = 1 | + (−10.9 + 2.49i)2-s + (−2.67 + 5.56i)3-s + (84.7 − 40.8i)4-s + (−8.95 − 39.2i)5-s + (15.4 − 67.5i)6-s + (52.7 + 25.4i)7-s + (−544. + 434. i)8-s + (127. + 160. i)9-s + (196. + 407. i)10-s + (−167. − 133. i)11-s + 580. i·12-s + (−305. + 382. i)13-s + (−640. − 146. i)14-s + (242. + 55.2i)15-s + (3.00e3 − 3.76e3i)16-s + 904. i·17-s + ⋯ |
| L(s) = 1 | + (−1.93 + 0.441i)2-s + (−0.171 + 0.356i)3-s + (2.64 − 1.27i)4-s + (−0.160 − 0.701i)5-s + (0.174 − 0.766i)6-s + (0.406 + 0.195i)7-s + (−3.01 + 2.40i)8-s + (0.525 + 0.659i)9-s + (0.620 + 1.28i)10-s + (−0.417 − 0.333i)11-s + 1.16i·12-s + (−0.500 + 0.628i)13-s + (−0.873 − 0.199i)14-s + (0.277 + 0.0634i)15-s + (2.93 − 3.67i)16-s + 0.758i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.310212 + 0.399453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.310212 + 0.399453i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-4.28e3 - 1.47e3i)T \) |
| good | 2 | \( 1 + (10.9 - 2.49i)T + (28.8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (2.67 - 5.56i)T + (-151. - 189. i)T^{2} \) |
| 5 | \( 1 + (8.95 + 39.2i)T + (-2.81e3 + 1.35e3i)T^{2} \) |
| 7 | \( 1 + (-52.7 - 25.4i)T + (1.04e4 + 1.31e4i)T^{2} \) |
| 11 | \( 1 + (167. + 133. i)T + (3.58e4 + 1.57e5i)T^{2} \) |
| 13 | \( 1 + (305. - 382. i)T + (-8.26e4 - 3.61e5i)T^{2} \) |
| 17 | \( 1 - 904. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + (-1.18e3 - 2.45e3i)T + (-1.54e6 + 1.93e6i)T^{2} \) |
| 23 | \( 1 + (581. - 2.54e3i)T + (-5.79e6 - 2.79e6i)T^{2} \) |
| 31 | \( 1 + (6.51e3 - 1.48e3i)T + (2.57e7 - 1.24e7i)T^{2} \) |
| 37 | \( 1 + (721. - 575. i)T + (1.54e7 - 6.76e7i)T^{2} \) |
| 41 | \( 1 + 3.39e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.49e4 - 3.41e3i)T + (1.32e8 + 6.37e7i)T^{2} \) |
| 47 | \( 1 + (-3.06e3 - 2.44e3i)T + (5.10e7 + 2.23e8i)T^{2} \) |
| 53 | \( 1 + (-756. - 3.31e3i)T + (-3.76e8 + 1.81e8i)T^{2} \) |
| 59 | \( 1 - 1.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (-30.4 + 63.3i)T + (-5.26e8 - 6.60e8i)T^{2} \) |
| 67 | \( 1 + (-1.02e4 - 1.28e4i)T + (-3.00e8 + 1.31e9i)T^{2} \) |
| 71 | \( 1 + (1.61e4 - 2.02e4i)T + (-4.01e8 - 1.75e9i)T^{2} \) |
| 73 | \( 1 + (6.11e4 + 1.39e4i)T + (1.86e9 + 8.99e8i)T^{2} \) |
| 79 | \( 1 + (-4.69e3 + 3.74e3i)T + (6.84e8 - 2.99e9i)T^{2} \) |
| 83 | \( 1 + (-5.57e4 + 2.68e4i)T + (2.45e9 - 3.07e9i)T^{2} \) |
| 89 | \( 1 + (7.60e3 - 1.73e3i)T + (5.03e9 - 2.42e9i)T^{2} \) |
| 97 | \( 1 + (3.97e4 + 8.25e4i)T + (-5.35e9 + 6.71e9i)T^{2} \) |
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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
−16.38250994114203062107927849811, −16.02661870812025159248208633791, −14.57596198887971628607383640452, −12.11806773989389887564843842889, −10.80793322121641990274366479370, −9.773791260149757761641567804513, −8.490600416439599237938124165623, −7.47832545255927004546567482097, −5.53495645895475783284235571445, −1.58446210010473499044710532183,
0.63213620858817282181473235481, 2.69900414289301634507670622130, 6.86247598028573485609136151556, 7.61135400117210928221182253703, 9.255946643920387156705896568560, 10.42904084227927159557094343508, 11.44729789022611581256694882096, 12.60876137771040216815018964205, 15.05632857166279096648816323107, 16.05135477218097907946790871479
