| L(s) = 1 | + (2.80 − 4.13i)2-s + (−2.32 + 4.64i)3-s + (−6.27 − 15.7i)4-s + (−2.11 + 4.56i)5-s + (12.6 + 22.6i)6-s + (8.76 + 31.5i)7-s + (−43.7 − 9.61i)8-s + (−16.1 − 21.6i)9-s + (12.9 + 21.5i)10-s + (24.3 + 23.0i)11-s + (87.7 + 7.49i)12-s + (−5.46 + 50.2i)13-s + (155. + 52.2i)14-s + (−16.2 − 20.4i)15-s + (−63.8 + 60.4i)16-s + (57.7 + 16.0i)17-s + ⋯ |
| L(s) = 1 | + (0.991 − 1.46i)2-s + (−0.447 + 0.894i)3-s + (−0.784 − 1.96i)4-s + (−0.188 + 0.407i)5-s + (0.863 + 1.54i)6-s + (0.473 + 1.70i)7-s + (−1.93 − 0.425i)8-s + (−0.598 − 0.800i)9-s + (0.409 + 0.680i)10-s + (0.667 + 0.632i)11-s + (2.11 + 0.180i)12-s + (−0.116 + 1.07i)13-s + (2.95 + 0.997i)14-s + (−0.280 − 0.351i)15-s + (−0.996 + 0.944i)16-s + (0.824 + 0.228i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.13185 + 0.241800i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13185 + 0.241800i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.32 - 4.64i)T \) |
| 59 | \( 1 + (-70.8 + 447. i)T \) |
| good | 2 | \( 1 + (-2.80 + 4.13i)T + (-2.96 - 7.43i)T^{2} \) |
| 5 | \( 1 + (2.11 - 4.56i)T + (-80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (-8.76 - 31.5i)T + (-293. + 176. i)T^{2} \) |
| 11 | \( 1 + (-24.3 - 23.0i)T + (72.0 + 1.32e3i)T^{2} \) |
| 13 | \( 1 + (5.46 - 50.2i)T + (-2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (-57.7 - 16.0i)T + (4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (82.6 + 62.8i)T + (1.83e3 + 6.60e3i)T^{2} \) |
| 23 | \( 1 + (-1.50 - 2.84i)T + (-6.82e3 + 1.00e4i)T^{2} \) |
| 29 | \( 1 + (76.1 - 51.6i)T + (9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (-73.2 - 96.3i)T + (-7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (-8.34 - 37.9i)T + (-4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (-180. - 95.7i)T + (3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (264. + 279. i)T + (-4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (175. - 81.3i)T + (6.72e4 - 7.91e4i)T^{2} \) |
| 53 | \( 1 + (126. - 210. i)T + (-6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (-488. - 331. i)T + (8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (33.2 - 150. i)T + (-2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (79.2 + 171. i)T + (-2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (-342. + 1.01e3i)T + (-3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (33.9 - 625. i)T + (-4.90e5 - 5.33e4i)T^{2} \) |
| 83 | \( 1 + (-233. + 1.42e3i)T + (-5.41e5 - 1.82e5i)T^{2} \) |
| 89 | \( 1 + (-866. - 1.27e3i)T + (-2.60e5 + 6.54e5i)T^{2} \) |
| 97 | \( 1 + (109. + 324. i)T + (-7.26e5 + 5.52e5i)T^{2} \) |
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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
−11.94437034046492363475564138174, −11.55308121335473905975694345774, −10.67091080142565499783363741978, −9.539215774149924623346027081954, −8.861814081556071971450390146932, −6.43459081086206073688137850929, −5.26774243529978046819282419329, −4.46601768160228938945551200710, −3.20561493743834924820958176218, −1.92044896154065252941114894513,
0.789078591944985745945553940758, 3.69221307151039604968637848414, 4.80292096114060557675239536855, 5.93274040604739906575452294946, 6.84531334502865359769432359921, 7.84841781073893630131291938836, 8.238896287201862741891943614904, 10.38605087963015388007666955635, 11.58067772081753721388042265152, 12.73033262460604221120319401072
