| L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.669 − 0.743i)3-s + (−0.809 + 0.587i)4-s + (0.5 + 0.866i)5-s + (−0.499 + 0.866i)6-s + (−0.296 − 2.81i)7-s + (0.809 + 0.587i)8-s + (−0.104 + 0.994i)9-s + (0.669 − 0.743i)10-s + (4.49 + 2.00i)11-s + (0.978 + 0.207i)12-s + (−1.96 + 0.417i)13-s + (−2.58 + 1.15i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (1.96 − 0.873i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.386 − 0.429i)3-s + (−0.404 + 0.293i)4-s + (0.223 + 0.387i)5-s + (−0.204 + 0.353i)6-s + (−0.111 − 1.06i)7-s + (0.286 + 0.207i)8-s + (−0.0348 + 0.331i)9-s + (0.211 − 0.235i)10-s + (1.35 + 0.604i)11-s + (0.282 + 0.0600i)12-s + (−0.544 + 0.115i)13-s + (−0.691 + 0.307i)14-s + (0.0797 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.475 − 0.211i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00103 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00103 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.901485 - 0.902419i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.901485 - 0.902419i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.669 + 0.743i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-4.43 - 3.36i)T \) |
| good | 7 | \( 1 + (0.296 + 2.81i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-4.49 - 2.00i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (1.96 - 0.417i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-1.96 + 0.873i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-4.09 - 0.869i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-0.0289 - 0.0210i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.51 + 4.67i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.60 + 2.78i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.11 + 3.46i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (6.96 + 1.47i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (-2.23 + 6.86i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.39 + 13.2i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-2.21 - 2.46i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 - 0.406T + 61T^{2} \) |
| 67 | \( 1 + (-2.22 - 3.85i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.08 + 10.3i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-3.58 - 1.59i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (0.705 - 0.314i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (6.12 - 6.79i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-4.61 + 3.35i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.01 + 1.46i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.932561893667487092005774537567, −9.384560532938878545662937794129, −8.117122492810892724047408653834, −7.16520929028906565791064670940, −6.74167292375165315797270651122, −5.44473901743306354985181248032, −4.30465606961512382143250815748, −3.43785662005531418731108041348, −2.02262841085117172480667558931, −0.857460652736834956911503930272,
1.18721182540732070393477446846, 2.99125284193077939566306431528, 4.26872976285087333711238316581, 5.28036375368034690708129630656, 5.91338964993117914499602583145, 6.66561666703647245669840940350, 7.83532268108781472405743352147, 8.778524268254969265818081444649, 9.352859598231722270947139577273, 9.911374972141366290307991981487
