| L(s) = 1 | + (−13.5 − 7.79i)3-s + (−175. + 101. i)5-s + (−284. − 191. i)7-s + (121.5 + 210. i)9-s + (437. − 758. i)11-s − 275. i·13-s + 3.15e3·15-s + (−3.79e3 − 2.19e3i)17-s + (1.16e4 − 6.75e3i)19-s + (2.34e3 + 4.80e3i)21-s + (−6.36e3 − 1.10e4i)23-s + (1.26e4 − 2.19e4i)25-s − 3.78e3i·27-s + 6.26e3·29-s + (1.76e4 + 1.02e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 − 0.288i)3-s + (−1.40 + 0.809i)5-s + (−0.828 − 0.559i)7-s + (0.166 + 0.288i)9-s + (0.328 − 0.569i)11-s − 0.125i·13-s + 0.935·15-s + (−0.773 − 0.446i)17-s + (1.70 − 0.984i)19-s + (0.252 + 0.519i)21-s + (−0.523 − 0.906i)23-s + (0.812 − 1.40i)25-s − 0.192i·27-s + 0.256·29-s + (0.593 + 0.342i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.005619450730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.005619450730\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (13.5 + 7.79i)T \) |
| 7 | \( 1 + (284. + 191. i)T \) |
| good | 5 | \( 1 + (175. - 101. i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-437. + 758. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 275. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (3.79e3 + 2.19e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-1.16e4 + 6.75e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (6.36e3 + 1.10e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 - 6.26e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.76e4 - 1.02e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (7.86e3 + 1.36e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + 6.99e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.13e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-4.01e4 + 2.31e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (6.40e4 - 1.10e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (2.42e5 + 1.40e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-8.47e4 + 4.89e4i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-8.72e4 + 1.51e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 3.45e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (1.04e5 + 6.03e4i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-3.81e5 - 6.61e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + 8.59e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (1.55e5 - 8.99e4i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 - 3.40e5iT - 8.32e11T^{2} \) |
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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
−10.17414526316776479387711061123, −8.932896359364812883988764088289, −7.72705118486550475231804600725, −7.01140862610025747000793678801, −6.35354536901271640298298142703, −4.80920278650357688290795401255, −3.66889364481441303208585648263, −2.84230023318574306257659455327, −0.73182632827635031099400961033, −0.00226816940048678435372441355,
1.35795253744171349214910023138, 3.28965565824746628506811153802, 4.16107150842552704744009813780, 5.15286323921058446431652091798, 6.27477917742128931119420680064, 7.42008356959333243257602813062, 8.334979966547029902293327450731, 9.356418361347569652706911185862, 10.07554799697211550985678506431, 11.56435711376520566482430604351
