| L(s) = 1 | + (−2.44 − 1.41i)2-s + (0.729 − 8.97i)3-s + (3.99 + 6.92i)4-s + (−30.1 − 17.4i)5-s + (−14.4 + 20.9i)6-s + (31.6 + 37.4i)7-s − 22.6i·8-s + (−79.9 − 13.0i)9-s + (49.2 + 85.3i)10-s + (−119. + 68.8i)11-s + (65.0 − 30.8i)12-s − 238.·13-s + (−24.5 − 136. i)14-s + (−178. + 258. i)15-s + (−32.0 + 55.4i)16-s + (198. − 114. i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.0810 − 0.996i)3-s + (0.249 + 0.433i)4-s + (−1.20 − 0.697i)5-s + (−0.402 + 0.581i)6-s + (0.645 + 0.763i)7-s − 0.353i·8-s + (−0.986 − 0.161i)9-s + (0.492 + 0.853i)10-s + (−0.985 + 0.569i)11-s + (0.451 − 0.214i)12-s − 1.41·13-s + (−0.125 − 0.695i)14-s + (−0.792 + 1.14i)15-s + (−0.125 + 0.216i)16-s + (0.685 − 0.396i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0687998 + 0.374517i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0687998 + 0.374517i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.44 + 1.41i)T \) |
| 3 | \( 1 + (-0.729 + 8.97i)T \) |
| 7 | \( 1 + (-31.6 - 37.4i)T \) |
| good | 5 | \( 1 + (30.1 + 17.4i)T + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (119. - 68.8i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + 238.T + 2.85e4T^{2} \) |
| 17 | \( 1 + (-198. + 114. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-258. + 448. i)T + (-6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (187. + 108. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + 565. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + (737. + 1.27e3i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-410. + 710. i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 - 747. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 984.T + 3.41e6T^{2} \) |
| 47 | \( 1 + (254. + 146. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (2.70e3 - 1.56e3i)T + (3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-3.25e3 + 1.87e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-1.13e3 + 1.96e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (1.37e3 + 2.38e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 5.47e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + (-1.24e3 - 2.15e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-2.96e3 + 5.14e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 3.54e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (-7.54e3 - 4.35e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 5.48e3T + 8.85e7T^{2} \) |
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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
−14.76122742805307884317010645425, −12.96151644158415164391247264372, −12.07966910918599380988142557281, −11.43853901459294766421217523853, −9.404439783411006923061452694493, −7.995201866861592264887229018213, −7.49825112353234431045657631542, −5.03830212479538276625827717169, −2.45471800991771623542290685951, −0.28738189062996641388264013086,
3.44439175109938820253440464072, 5.17152098365307538078048617565, 7.44543453531146282762761890417, 8.185638600557708470808135506762, 10.05705598537602063472418284218, 10.77159512766180842837278877596, 11.90596123369489522243507036442, 14.27598222431946619667498685428, 14.85450644292350229286759707646, 16.01332211224535730435152454635
