| L(s) = 1 | + (0.993 + 0.478i)2-s + (−0.0482 + 0.644i)3-s + (−0.488 − 0.612i)4-s + (−3.17 − 0.979i)5-s + (−0.356 + 0.616i)6-s + (1.23 + 2.13i)7-s + (−0.683 − 2.99i)8-s + (2.55 + 0.384i)9-s + (−2.68 − 2.49i)10-s + (0.748 − 0.937i)11-s + (0.418 − 0.285i)12-s + (−4.22 + 3.91i)13-s + (0.203 + 2.71i)14-s + (0.784 − 1.99i)15-s + (0.404 − 1.77i)16-s + (1.02 − 0.314i)17-s + ⋯ |
| L(s) = 1 | + (0.702 + 0.338i)2-s + (−0.0278 + 0.371i)3-s + (−0.244 − 0.306i)4-s + (−1.41 − 0.437i)5-s + (−0.145 + 0.251i)6-s + (0.465 + 0.807i)7-s + (−0.241 − 1.05i)8-s + (0.851 + 0.128i)9-s + (−0.849 − 0.788i)10-s + (0.225 − 0.282i)11-s + (0.120 − 0.0823i)12-s + (−1.17 + 1.08i)13-s + (0.0542 + 0.724i)14-s + (0.202 − 0.515i)15-s + (0.101 − 0.442i)16-s + (0.247 − 0.0763i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.882537 + 0.159099i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.882537 + 0.159099i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (2.45 + 6.08i)T \) |
| good | 2 | \( 1 + (-0.993 - 0.478i)T + (1.24 + 1.56i)T^{2} \) |
| 3 | \( 1 + (0.0482 - 0.644i)T + (-2.96 - 0.447i)T^{2} \) |
| 5 | \( 1 + (3.17 + 0.979i)T + (4.13 + 2.81i)T^{2} \) |
| 7 | \( 1 + (-1.23 - 2.13i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.748 + 0.937i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (4.22 - 3.91i)T + (0.971 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.02 + 0.314i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-2.13 + 0.321i)T + (18.1 - 5.60i)T^{2} \) |
| 23 | \( 1 + (2.16 + 5.50i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (-0.177 - 2.37i)T + (-28.6 + 4.32i)T^{2} \) |
| 31 | \( 1 + (-6.31 + 4.30i)T + (11.3 - 28.8i)T^{2} \) |
| 37 | \( 1 + (2.83 - 4.90i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.77 + 4.22i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (-3.30 - 4.14i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-2.69 - 2.50i)T + (3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (-0.208 + 0.914i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (2.31 + 1.57i)T + (22.2 + 56.7i)T^{2} \) |
| 67 | \( 1 + (-11.8 + 1.78i)T + (64.0 - 19.7i)T^{2} \) |
| 71 | \( 1 + (5.54 - 14.1i)T + (-52.0 - 48.2i)T^{2} \) |
| 73 | \( 1 + (4.99 - 4.63i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (1.18 + 2.05i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.533 - 7.11i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (0.113 - 1.51i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + (6.73 - 8.44i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−15.70525618997617741537877993065, −15.10630382801623908956626801524, −14.02444045653120162285715066052, −12.42321461776912538179371393555, −11.75892552234353425131369422470, −9.949245027890213825257009428564, −8.598481195985574487263055552796, −7.01691791649351277509195021052, −5.02116596424919735914380159025, −4.14847543939525586155832743892,
3.50638318597277095651229242111, 4.72765157710429876239442766895, 7.34492575041471470739952230894, 7.935630213240924471869940932621, 10.19018922086412880174489790580, 11.67200370311089545013438426045, 12.30624625935540003129003199618, 13.50107169342892283080034547592, 14.69400915501883701229240746908, 15.63390677859709095263942171346
