| L(s) = 1 | + (−1.73 − i)2-s + (4.48 + 2.62i)3-s + (1.99 + 3.46i)4-s + (−5.27 + 9.13i)5-s + (−5.13 − 9.03i)6-s + (17.7 + 5.44i)7-s − 7.99i·8-s + (13.1 + 23.5i)9-s + (18.2 − 10.5i)10-s + (26.6 − 15.4i)11-s + (−0.141 + 20.7i)12-s + 19.8i·13-s + (−25.2 − 27.1i)14-s + (−47.6 + 27.0i)15-s + (−8 + 13.8i)16-s + (−46.3 − 80.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.862 + 0.505i)3-s + (0.249 + 0.433i)4-s + (−0.471 + 0.816i)5-s + (−0.349 − 0.614i)6-s + (0.955 + 0.293i)7-s − 0.353i·8-s + (0.488 + 0.872i)9-s + (0.577 − 0.333i)10-s + (0.731 − 0.422i)11-s + (−0.00341 + 0.499i)12-s + 0.423i·13-s + (−0.481 − 0.517i)14-s + (−0.820 + 0.466i)15-s + (−0.125 + 0.216i)16-s + (−0.661 − 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.21870 + 0.387305i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21870 + 0.387305i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.73 + i)T \) |
| 3 | \( 1 + (-4.48 - 2.62i)T \) |
| 7 | \( 1 + (-17.7 - 5.44i)T \) |
| good | 5 | \( 1 + (5.27 - 9.13i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-26.6 + 15.4i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 19.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (46.3 + 80.2i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (118. + 68.6i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-37.6 - 21.7i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 134. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-144. + 83.6i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-191. + 332. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 107.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (120. - 209. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-432. + 249. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-366. - 634. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (265. + 153. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (280. + 485. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 74.2iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-141. + 81.6i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (437. - 757. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 406.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (526. - 911. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 243. iT - 9.12e5T^{2} \) |
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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
−15.43316602344757819442730132356, −14.71282904740792198053142269186, −13.51758027279188214341795372348, −11.55798614425755721486552278166, −10.90481008753804278169901232101, −9.339474995662080822475881695594, −8.367622064925491405647527375556, −7.03356654267477649097927870399, −4.27778485419507690254107983749, −2.49783835648320092326696177864,
1.50511377620774824331762936665, 4.35828057829564904324543467280, 6.68221236596946648353272900730, 8.242482755408048834835746487455, 8.593969003214095558177949901858, 10.32766508672875581183729053937, 11.97101960957801444089459650001, 13.09549909536159280425053761314, 14.60823234041104175836897266881, 15.18051289081669226511742527213
