| L(s) = 1 | + (6.30 − 4.05i)2-s + (−1.28 + 8.90i)3-s + (10.0 − 21.9i)4-s + (−78.9 + 23.1i)5-s + (28.0 + 61.3i)6-s + (−51.3 − 59.2i)7-s + (8.35 + 58.0i)8-s + (−77.7 − 22.8i)9-s + (−403. + 466. i)10-s + (−206. − 132. i)11-s + (182. + 117. i)12-s + (−697. + 804. i)13-s + (−563. − 165. i)14-s + (−105. − 732. i)15-s + (794. + 916. i)16-s + (97.3 + 213. i)17-s + ⋯ |
| L(s) = 1 | + (1.11 − 0.716i)2-s + (−0.0821 + 0.571i)3-s + (0.313 − 0.687i)4-s + (−1.41 + 0.414i)5-s + (0.317 + 0.695i)6-s + (−0.395 − 0.456i)7-s + (0.0461 + 0.320i)8-s + (−0.319 − 0.0939i)9-s + (−1.27 + 1.47i)10-s + (−0.514 − 0.330i)11-s + (0.366 + 0.235i)12-s + (−1.14 + 1.32i)13-s + (−0.768 − 0.225i)14-s + (−0.120 − 0.841i)15-s + (0.775 + 0.895i)16-s + (0.0817 + 0.178i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.331054 + 0.701860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.331054 + 0.701860i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.28 - 8.90i)T \) |
| 23 | \( 1 + (1.09e3 + 2.29e3i)T \) |
| good | 2 | \( 1 + (-6.30 + 4.05i)T + (13.2 - 29.1i)T^{2} \) |
| 5 | \( 1 + (78.9 - 23.1i)T + (2.62e3 - 1.68e3i)T^{2} \) |
| 7 | \( 1 + (51.3 + 59.2i)T + (-2.39e3 + 1.66e4i)T^{2} \) |
| 11 | \( 1 + (206. + 132. i)T + (6.69e4 + 1.46e5i)T^{2} \) |
| 13 | \( 1 + (697. - 804. i)T + (-5.28e4 - 3.67e5i)T^{2} \) |
| 17 | \( 1 + (-97.3 - 213. i)T + (-9.29e5 + 1.07e6i)T^{2} \) |
| 19 | \( 1 + (-373. + 816. i)T + (-1.62e6 - 1.87e6i)T^{2} \) |
| 29 | \( 1 + (-2.22e3 - 4.88e3i)T + (-1.34e7 + 1.55e7i)T^{2} \) |
| 31 | \( 1 + (207. + 1.44e3i)T + (-2.74e7 + 8.06e6i)T^{2} \) |
| 37 | \( 1 + (-1.01e4 - 2.97e3i)T + (5.83e7 + 3.74e7i)T^{2} \) |
| 41 | \( 1 + (1.37e4 - 4.04e3i)T + (9.74e7 - 6.26e7i)T^{2} \) |
| 43 | \( 1 + (349. - 2.42e3i)T + (-1.41e8 - 4.14e7i)T^{2} \) |
| 47 | \( 1 + 1.46e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.04e4 + 1.20e4i)T + (-5.95e7 + 4.13e8i)T^{2} \) |
| 59 | \( 1 + (2.98e3 - 3.43e3i)T + (-1.01e8 - 7.07e8i)T^{2} \) |
| 61 | \( 1 + (-3.42e3 - 2.38e4i)T + (-8.10e8 + 2.37e8i)T^{2} \) |
| 67 | \( 1 + (-4.81e4 + 3.09e4i)T + (5.60e8 - 1.22e9i)T^{2} \) |
| 71 | \( 1 + (1.85e4 - 1.19e4i)T + (7.49e8 - 1.64e9i)T^{2} \) |
| 73 | \( 1 + (3.46e4 - 7.57e4i)T + (-1.35e9 - 1.56e9i)T^{2} \) |
| 79 | \( 1 + (6.48e4 - 7.48e4i)T + (-4.37e8 - 3.04e9i)T^{2} \) |
| 83 | \( 1 + (7.11e4 + 2.09e4i)T + (3.31e9 + 2.12e9i)T^{2} \) |
| 89 | \( 1 + (1.68e4 - 1.17e5i)T + (-5.35e9 - 1.57e9i)T^{2} \) |
| 97 | \( 1 + (-1.38e5 + 4.05e4i)T + (7.22e9 - 4.64e9i)T^{2} \) |
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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
−14.20878261625960676971506135651, −12.89571054568628112325363480791, −11.81263508061803867493397923986, −11.22986755953258259649227083040, −10.05886654080142460400308950461, −8.287142135377293614087787038913, −6.85594176387039728083889736398, −4.85428844483031857188148846719, −3.98350526755961944853801617414, −2.84105926257265273976088782768,
0.23616762893250517704083757336, 3.18092852398274619188412017493, 4.70527664727944402629217385961, 5.81557571852156635456621283395, 7.38661242148940469763806125437, 7.987033359432590041976309739808, 9.935801737194095442177287743972, 11.81439591501428232958906640597, 12.46108895399699299000799947095, 13.21453989443053544407603081536
