| L(s) = 1 | + (−0.451 + 1.34i)2-s + (−1.87 + 2.34i)3-s + (−1.59 − 1.21i)4-s + (−7.13 − 2.84i)5-s + (−2.29 − 3.57i)6-s + (1.14 − 0.531i)7-s + (2.34 − 1.58i)8-s + (−1.96 − 8.78i)9-s + (7.03 − 8.27i)10-s + (3.29 + 0.915i)11-s + (5.82 − 1.45i)12-s + (3.53 − 6.66i)13-s + (0.193 + 1.78i)14-s + (20.0 − 11.3i)15-s + (1.07 + 3.85i)16-s + (−9.75 + 21.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 + 0.670i)2-s + (−0.625 + 0.780i)3-s + (−0.398 − 0.302i)4-s + (−1.42 − 0.568i)5-s + (−0.381 − 0.595i)6-s + (0.164 − 0.0759i)7-s + (0.292 − 0.198i)8-s + (−0.218 − 0.975i)9-s + (0.703 − 0.827i)10-s + (0.299 + 0.0832i)11-s + (0.485 − 0.121i)12-s + (0.271 − 0.512i)13-s + (0.0138 + 0.127i)14-s + (1.33 − 0.758i)15-s + (0.0668 + 0.240i)16-s + (−0.573 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.679538 + 0.330368i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.679538 + 0.330368i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.451 - 1.34i)T \) |
| 3 | \( 1 + (1.87 - 2.34i)T \) |
| 59 | \( 1 + (24.5 + 53.6i)T \) |
| good | 5 | \( 1 + (7.13 + 2.84i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (-1.14 + 0.531i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (-3.29 - 0.915i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (-3.53 + 6.66i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (9.75 - 21.0i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (-9.04 + 1.99i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (-1.75 - 0.288i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (1.28 + 3.80i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (-8.69 - 1.91i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (-10.1 + 15.0i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (-21.7 + 3.56i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (-3.51 - 12.6i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (-78.8 + 31.4i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (3.42 - 2.90i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (-92.3 - 31.1i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-15.6 - 23.0i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (-61.6 + 24.5i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (-115. + 12.6i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (2.13 - 1.28i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (-75.6 + 4.10i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (23.1 + 68.7i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (-151. - 16.4i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−11.25975013129805940927764038999, −10.57151267974434886100519849223, −9.372592069522917044228002541969, −8.516293894417146557141866716898, −7.74413772604873200348592913046, −6.55311161173168869369090271986, −5.44490752088918299046937591607, −4.39619395898114225045469012583, −3.70822291863362767560923218583, −0.69403829559272147896082634079,
0.76299620551036027839400556458, 2.52261877416141729405220683158, 3.86458416606858017318898452540, 5.00933254060319345316699581900, 6.57939592549551015277821613939, 7.38264704503808453065329682898, 8.155643147781450914444758332666, 9.263854894368656739792594505759, 10.63988753849439602013508563330, 11.42601319860398408016313235275
