| L(s) = 1 | + (0.418 + 0.915i)2-s + (0.166 − 0.0488i)3-s + (0.646 − 0.745i)4-s + (0.841 + 0.540i)5-s + (0.114 + 0.131i)6-s + (−0.333 − 2.31i)7-s + (2.88 + 0.847i)8-s + (−2.49 + 1.60i)9-s + (−0.143 + 0.996i)10-s + (−2.29 + 5.02i)11-s + (0.0710 − 0.155i)12-s + (0.372 − 2.59i)13-s + (1.98 − 1.27i)14-s + (0.166 + 0.0488i)15-s + (0.149 + 1.04i)16-s + (−3.91 − 4.51i)17-s + ⋯ |
| L(s) = 1 | + (0.295 + 0.647i)2-s + (0.0960 − 0.0281i)3-s + (0.323 − 0.372i)4-s + (0.376 + 0.241i)5-s + (0.0466 + 0.0538i)6-s + (−0.125 − 0.876i)7-s + (1.01 + 0.299i)8-s + (−0.832 + 0.535i)9-s + (−0.0453 + 0.315i)10-s + (−0.691 + 1.51i)11-s + (0.0205 − 0.0449i)12-s + (0.103 − 0.719i)13-s + (0.530 − 0.340i)14-s + (0.0429 + 0.0126i)15-s + (0.0374 + 0.260i)16-s + (−0.948 − 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28763 + 0.346154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28763 + 0.346154i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (-3.97 - 2.67i)T \) |
| good | 2 | \( 1 + (-0.418 - 0.915i)T + (-1.30 + 1.51i)T^{2} \) |
| 3 | \( 1 + (-0.166 + 0.0488i)T + (2.52 - 1.62i)T^{2} \) |
| 7 | \( 1 + (0.333 + 2.31i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (2.29 - 5.02i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.372 + 2.59i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (3.91 + 4.51i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (1.00 - 1.16i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (6.49 + 7.49i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (1.90 + 0.558i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.981 + 0.630i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-9.20 - 5.91i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (10.4 - 3.08i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 5.54T + 47T^{2} \) |
| 53 | \( 1 + (-0.548 - 3.81i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (1.36 - 9.49i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-7.47 - 2.19i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-0.387 - 0.847i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (1.24 + 2.73i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-6.99 + 8.06i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.115 + 0.805i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-4.13 + 2.65i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (6.73 - 1.97i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (5.84 + 3.75i)T + (40.2 + 88.2i)T^{2} \) |
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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
−13.64462072987569246848995886305, −13.15123716255985629269391051673, −11.35640172883775483102769510013, −10.54494453711905803595391380392, −9.576782247275180351648292518299, −7.74201508202124560053776923850, −7.11369679960574727423462917944, −5.74005719279066532800966783682, −4.65545573329759393584937019482, −2.38613048864726234685241579795,
2.35880305341988871534911279487, 3.59837176348683258728198397470, 5.46062040248650886634920469012, 6.62092079053191422539695042053, 8.450562065147446531736141265751, 9.011648570940081308295214747252, 10.82141112465756620892610079455, 11.31969148021079744239505440682, 12.53833570830991044286754078544, 13.19214364287450085853693013564
