L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.450 + 0.779i)5-s + (−2.62 − 0.296i)7-s − 0.999i·8-s + (0.779 + 0.450i)10-s + (−2.70 − 1.56i)11-s − 2.29i·13-s + (−2.42 + 1.05i)14-s + (−0.5 − 0.866i)16-s + (2.57 − 4.46i)17-s + (2.38 − 1.37i)19-s + 0.900·20-s − 3.12·22-s + (−1.48 + 0.857i)23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.201 + 0.348i)5-s + (−0.993 − 0.112i)7-s − 0.353i·8-s + (0.246 + 0.142i)10-s + (−0.816 − 0.471i)11-s − 0.637i·13-s + (−0.648 + 0.282i)14-s + (−0.125 − 0.216i)16-s + (0.624 − 1.08i)17-s + (0.546 − 0.315i)19-s + 0.201·20-s − 0.666·22-s + (−0.309 + 0.178i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.612496184\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.612496184\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.62 + 0.296i)T \) |
good | 5 | \( 1 + (-0.450 - 0.779i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.70 + 1.56i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.29iT - 13T^{2} \) |
| 17 | \( 1 + (-2.57 + 4.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.38 + 1.37i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.48 - 0.857i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.14iT - 29T^{2} \) |
| 31 | \( 1 + (8.66 + 5.00i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.73 + 8.20i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 - 0.546T + 43T^{2} \) |
| 47 | \( 1 + (-3.93 - 6.80i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-12.0 - 6.97i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.99 + 6.91i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.28 - 3.62i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.83 - 3.17i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (10.9 + 6.30i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.27 - 5.67i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.368T + 83T^{2} \) |
| 89 | \( 1 + (6.00 + 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10.2iT - 97T^{2} \) |
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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
−9.741259927841432134158453909525, −8.952780666251753766129729825372, −7.61477115340852376775612444464, −7.07358712923906120822873211348, −5.79752404335221283183794960601, −5.52370490206335629590936618343, −4.13061619961559064633220477550, −3.11354808998481474429968216383, −2.50746173863055204990334181303, −0.55097140676230452440329169381,
1.76703480821193348239996434520, 3.11766521277367553024933763863, 3.92302190667863973191657356313, 5.15416162332411380216659341377, 5.70982389755336639664348091219, 6.74707878490074682143338203472, 7.38336185633398085074446134693, 8.442922552033291447411521812741, 9.206425215636960151544062294430, 10.12794580574648473236675061707