| L(s) = 1 | − i·2-s + 8·3-s + 7·4-s − 8i·6-s + 14·7-s − 15i·8-s + 37·9-s − 20i·11-s + 56·12-s − 58i·13-s − 14i·14-s + 41·16-s + (−68 + 17i)17-s − 37i·18-s − 80·19-s + ⋯ |
| L(s) = 1 | − 0.353i·2-s + 1.53·3-s + 0.875·4-s − 0.544i·6-s + 0.755·7-s − 0.662i·8-s + 1.37·9-s − 0.548i·11-s + 1.34·12-s − 1.23i·13-s − 0.267i·14-s + 0.640·16-s + (−0.970 + 0.242i)17-s − 0.484i·18-s − 0.965·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.364563402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.364563402\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + (68 - 17i)T \) |
| good | 2 | \( 1 + iT - 8T^{2} \) |
| 3 | \( 1 - 8T + 27T^{2} \) |
| 7 | \( 1 - 14T + 343T^{2} \) |
| 11 | \( 1 + 20iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 58iT - 2.19e3T^{2} \) |
| 19 | \( 1 + 80T + 6.85e3T^{2} \) |
| 23 | \( 1 - 118T + 1.21e4T^{2} \) |
| 29 | \( 1 - 126iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 70iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 134T + 5.06e4T^{2} \) |
| 41 | \( 1 - 100iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 272iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 464iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 642iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 180T + 2.05e5T^{2} \) |
| 61 | \( 1 + 110iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 924iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 90iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 828T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.33e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 552iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.37e3T + 9.12e5T^{2} \) |
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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
−10.83332764987046128352844403802, −9.686093146921244222301113634369, −8.645453166227106333881918707611, −8.048677763764230405898927565495, −7.19726739581177843532159752185, −5.99660369516687135390217843673, −4.43533566661621721496821003341, −3.17824105483652008526927068339, −2.50963361618824315808548965487, −1.30530853611715698700040911666,
1.87856241144005307199043095477, 2.37652648658096049057168116665, 3.84509133227563489834215382424, 4.93799221350388087564441150223, 6.62336266089388409498454841080, 7.18998965101606303987653175214, 8.227462043810885335529392441679, 8.784020859390785041390781401899, 9.760474019427137677442984441526, 10.94826705527549789951457968561
