| L(s) = 1 | + (−2.73 + 4.73i)2-s + (58.8 − 33.9i)3-s + (113. + 195. i)4-s + (586. + 338. i)5-s + 372. i·6-s + (−168. − 2.39e3i)7-s − 2.63e3·8-s + (−968. + 1.67e3i)9-s + (−3.20e3 + 1.85e3i)10-s + (−7.99e3 − 1.38e4i)11-s + (1.33e4 + 7.68e3i)12-s + 7.14e3i·13-s + (1.18e4 + 5.75e3i)14-s + 4.60e4·15-s + (−2.17e4 + 3.76e4i)16-s + (3.76e4 − 2.17e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.170 + 0.296i)2-s + (0.726 − 0.419i)3-s + (0.441 + 0.764i)4-s + (0.938 + 0.541i)5-s + 0.287i·6-s + (−0.0702 − 0.997i)7-s − 0.643·8-s + (−0.147 + 0.255i)9-s + (−0.320 + 0.185i)10-s + (−0.545 − 0.945i)11-s + (0.641 + 0.370i)12-s + 0.249i·13-s + (0.307 + 0.149i)14-s + 0.909·15-s + (−0.331 + 0.574i)16-s + (0.451 − 0.260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.66373 + 0.395831i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66373 + 0.395831i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (168. + 2.39e3i)T \) |
| good | 2 | \( 1 + (2.73 - 4.73i)T + (-128 - 221. i)T^{2} \) |
| 3 | \( 1 + (-58.8 + 33.9i)T + (3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-586. - 338. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (7.99e3 + 1.38e4i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 7.14e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-3.76e4 + 2.17e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.66e5 + 9.61e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-9.19e4 + 1.59e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.23e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + (9.28e5 - 5.35e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (1.10e6 - 1.91e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 - 1.82e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.38e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-2.72e6 - 1.57e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (2.66e6 + 4.61e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.22e7 + 7.07e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-1.39e7 - 8.02e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.45e6 - 2.51e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 1.93e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (1.68e7 - 9.75e6i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-2.39e7 + 4.15e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 8.03e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (2.83e7 + 1.63e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 6.94e7iT - 7.83e15T^{2} \) |
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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
−20.76871326237813070677438251127, −19.22824840075434039104090226566, −17.62202489814855778885258044922, −16.37844171164198437761659958079, −14.24714223017362443366476791264, −13.17086579812246010962775283471, −10.73341196285223182500525074982, −8.389104704977813147315158388247, −6.77343369109864426273231715885, −2.74307320132641625332301878977,
2.19748764202604919572738127912, 5.71868867167670843441578623741, 8.983972764521192289896346944374, 10.13534056126251744352177520890, 12.46480292173377020279464813282, 14.52819402631070833988500686561, 15.56412024035360177885987809542, 17.71022995151110726581990577424, 19.25762210512032907789433135958, 20.67240018798615021274415236303
