| L(s) = 1 | + (0.194 − 2.59i)2-s + (−1.61 + 1.10i)3-s + (−4.69 − 0.708i)4-s + (−0.618 + 2.71i)5-s + (2.53 + 4.39i)6-s + (2.38 − 1.13i)7-s + (−1.59 + 6.96i)8-s + (0.298 − 0.760i)9-s + (6.90 + 2.12i)10-s + (1.50 − 3.84i)11-s + (8.36 − 4.02i)12-s + (3.69 + 3.42i)13-s + (−2.48 − 6.41i)14-s + (−1.98 − 5.05i)15-s + (8.66 + 2.67i)16-s + (0.868 + 3.80i)17-s + ⋯ |
| L(s) = 1 | + (0.137 − 1.83i)2-s + (−0.931 + 0.635i)3-s + (−2.34 − 0.354i)4-s + (−0.276 + 1.21i)5-s + (1.03 + 1.79i)6-s + (0.902 − 0.429i)7-s + (−0.562 + 2.46i)8-s + (0.0994 − 0.253i)9-s + (2.18 + 0.673i)10-s + (0.454 − 1.15i)11-s + (2.41 − 1.16i)12-s + (1.02 + 0.949i)13-s + (−0.663 − 1.71i)14-s + (−0.512 − 1.30i)15-s + (2.16 + 0.668i)16-s + (0.210 + 0.922i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.862050 - 0.277941i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.862050 - 0.277941i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.38 + 1.13i)T \) |
| 43 | \( 1 + (-6.51 - 0.728i)T \) |
| good | 2 | \( 1 + (-0.194 + 2.59i)T + (-1.97 - 0.298i)T^{2} \) |
| 3 | \( 1 + (1.61 - 1.10i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (0.618 - 2.71i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (-1.50 + 3.84i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-3.69 - 3.42i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.868 - 3.80i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + (0.570 - 0.715i)T + (-4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (-4.36 - 5.47i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (0.0729 - 0.973i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (0.401 - 5.35i)T + (-30.6 - 4.62i)T^{2} \) |
| 37 | \( 1 - 1.87T + 37T^{2} \) |
| 41 | \( 1 + (5.83 - 2.80i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (0.492 + 1.25i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (10.0 + 3.10i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.594 + 0.551i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (0.314 + 4.19i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (2.13 + 5.43i)T + (-49.1 + 45.5i)T^{2} \) |
| 71 | \( 1 + (-5.44 - 13.8i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (-2.63 + 11.5i)T + (-65.7 - 31.6i)T^{2} \) |
| 79 | \( 1 + (-2.11 - 3.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.29 + 17.2i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (-8.27 - 3.98i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (-3.57 - 4.47i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−11.28827535122176344345579630123, −10.91457427477863120122264290121, −10.52054080029046144621917647537, −9.256982800665749616783684359778, −8.177038504077638450774420173180, −6.44040187943579791194067603668, −5.20238910379573215427437746741, −4.02335025182607153703840189507, −3.32255382960264711819061788335, −1.47145482726457325673785885045,
0.863386621773838751931910545132, 4.40295718603602815719698282748, 5.10874657057570160702114590536, 5.87638690648178697407265810955, 6.90137761884821375755615773708, 7.81041059061748910821657063879, 8.613034764390856735470319288800, 9.377002613596167827398989410542, 11.13180145514483921638481890232, 12.30100701676993565863942465125
