L(s) = 1 | + (4.33 + 2.5i)2-s + (3.5 − 6.06i)3-s + (8.50 + 14.7i)4-s − 7i·5-s + (30.3 − 17.5i)6-s + (11.2 − 6.5i)7-s + 45.0i·8-s + (−11 − 19.0i)9-s + (17.5 − 30.3i)10-s + (−22.5 − 13i)11-s + 119.·12-s + 65·14-s + (−42.4 − 24.5i)15-s + (−44.5 + 77.0i)16-s + (38.5 + 66.6i)17-s − 109. i·18-s + ⋯ |
L(s) = 1 | + (1.53 + 0.883i)2-s + (0.673 − 1.16i)3-s + (1.06 + 1.84i)4-s − 0.626i·5-s + (2.06 − 1.19i)6-s + (0.607 − 0.350i)7-s + 1.98i·8-s + (−0.407 − 0.705i)9-s + (0.553 − 0.958i)10-s + (−0.617 − 0.356i)11-s + 2.86·12-s + 1.24·14-s + (−0.730 − 0.421i)15-s + (−0.695 + 1.20i)16-s + (0.549 + 0.951i)17-s − 1.44i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.81553 + 0.186038i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.81553 + 0.186038i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (-4.33 - 2.5i)T + (4 + 6.92i)T^{2} \) |
| 3 | \( 1 + (-3.5 + 6.06i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + 7iT - 125T^{2} \) |
| 7 | \( 1 + (-11.2 + 6.5i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (22.5 + 13i)T + (665.5 + 1.15e3i)T^{2} \) |
| 17 | \( 1 + (-38.5 - 66.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (109. - 63i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (48 - 83.1i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-41 + 71.0i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 196iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (113. + 65.5i)T + (2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (290. + 168i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (100.5 + 174. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 - 105iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 432T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-254. + 147i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-28 - 48.4i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (413. + 239i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-7.79 + 4.5i)T + (1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + 98iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.30e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 308iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (1.03e3 + 595i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-60.6 + 35i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.63764618508438306435229718904, −12.16865727561490580933157025837, −10.62382498500070411694525986055, −8.415492744551032383546926256235, −8.018516461547173647722718065182, −6.97233687375930340322332379351, −5.89355668706603221746918772156, −4.74722881677669615222018966773, −3.42481092976998314973810027226, −1.77637518177434355787469299384,
2.33876007739535327754470567914, 3.18323851303329639418922070096, 4.45948895997553759531399212611, 5.10471637257535172190693197088, 6.63442220005653328332266797979, 8.389842785634090846472795865166, 9.779805775072412023082924929511, 10.55341626655742030578000630308, 11.28499665210429731126773327805, 12.33776345822066352368574794571