| L(s) = 1 | + (1.30 + 2.16i)2-s + (−2.33 + 1.88i)3-s + (−1.12 + 2.12i)4-s + (0.380 + 3.50i)5-s + (−7.13 − 2.59i)6-s + (−11.3 + 3.83i)7-s + (4.04 − 0.219i)8-s + (1.87 − 8.80i)9-s + (−7.09 + 5.39i)10-s + (5.10 + 3.45i)11-s + (−1.38 − 7.06i)12-s + (−12.9 − 12.2i)13-s + (−23.1 − 19.6i)14-s + (−7.49 − 7.44i)15-s + (11.1 + 16.4i)16-s + (−8.29 + 24.6i)17-s + ⋯ |
| L(s) = 1 | + (0.652 + 1.08i)2-s + (−0.777 + 0.629i)3-s + (−0.281 + 0.530i)4-s + (0.0761 + 0.700i)5-s + (−1.18 − 0.432i)6-s + (−1.62 + 0.548i)7-s + (0.505 − 0.0273i)8-s + (0.208 − 0.978i)9-s + (−0.709 + 0.539i)10-s + (0.463 + 0.314i)11-s + (−0.115 − 0.588i)12-s + (−0.992 − 0.940i)13-s + (−1.65 − 1.40i)14-s + (−0.499 − 0.496i)15-s + (0.695 + 1.02i)16-s + (−0.487 + 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.230643 - 1.08486i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.230643 - 1.08486i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.33 - 1.88i)T \) |
| 59 | \( 1 + (-3.85 - 58.8i)T \) |
| good | 2 | \( 1 + (-1.30 - 2.16i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (-0.380 - 3.50i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (11.3 - 3.83i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (-5.10 - 3.45i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (12.9 + 12.2i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (8.29 - 24.6i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (1.18 + 7.24i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (4.36 - 1.21i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (0.602 - 1.00i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (8.00 - 48.8i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (-1.62 + 30.0i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (-17.9 - 4.97i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (-42.8 - 63.1i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (4.67 - 42.9i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (28.3 - 37.2i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (21.1 - 12.7i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (3.80 + 70.1i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (-0.880 + 8.09i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (-31.5 + 37.1i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (3.46 - 8.70i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (21.0 + 45.4i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (-89.4 + 148. i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (-52.1 - 61.3i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−12.76752415390149252732497075730, −12.53981329026186766200287837081, −10.80608718210821544273400109541, −10.17308705342387548792138683080, −9.099550720339442900602944741302, −7.30166084851045708520502000714, −6.37193247084981544500126776413, −5.90165500408911481311830573982, −4.56077987744528049361970377730, −3.19290552149893777098189385607,
0.57981421179920001292334413878, 2.38456586425430235044610860766, 3.97500200831139970496059139877, 5.10322395868818704809633548290, 6.55123165072132904435425692576, 7.40132287038346503694940389417, 9.301565371672138945150780887262, 10.09904068091741236158564069329, 11.32280343612995616334677367660, 12.03000141936595681871334440065
