| L(s) = 1 | + (0.671 − 0.106i)2-s + (−1.03 − 0.841i)3-s + (−1.46 + 0.475i)4-s + (0.835 − 2.07i)5-s + (−0.788 − 0.455i)6-s + (3.55 − 2.30i)7-s + (−2.14 + 1.09i)8-s + (−0.251 − 1.18i)9-s + (0.340 − 1.48i)10-s + (−1.45 − 1.31i)11-s + (1.92 + 0.737i)12-s + (−0.0419 + 0.0161i)13-s + (2.13 − 1.92i)14-s + (−2.61 + 1.45i)15-s + (1.16 − 0.845i)16-s + (−0.125 + 2.38i)17-s + ⋯ |
| L(s) = 1 | + (0.475 − 0.0752i)2-s + (−0.600 − 0.486i)3-s + (−0.731 + 0.237i)4-s + (0.373 − 0.927i)5-s + (−0.321 − 0.185i)6-s + (1.34 − 0.871i)7-s + (−0.758 + 0.386i)8-s + (−0.0838 − 0.394i)9-s + (0.107 − 0.468i)10-s + (−0.439 − 0.395i)11-s + (0.554 + 0.212i)12-s + (−0.0116 + 0.00447i)13-s + (0.571 − 0.514i)14-s + (−0.675 + 0.375i)15-s + (0.290 − 0.211i)16-s + (−0.0303 + 0.578i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.853139 - 0.695108i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.853139 - 0.695108i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.835 + 2.07i)T \) |
| 31 | \( 1 + (-4.86 + 2.71i)T \) |
| good | 2 | \( 1 + (-0.671 + 0.106i)T + (1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (1.03 + 0.841i)T + (0.623 + 2.93i)T^{2} \) |
| 7 | \( 1 + (-3.55 + 2.30i)T + (2.84 - 6.39i)T^{2} \) |
| 11 | \( 1 + (1.45 + 1.31i)T + (1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.0419 - 0.0161i)T + (9.66 - 8.69i)T^{2} \) |
| 17 | \( 1 + (0.125 - 2.38i)T + (-16.9 - 1.77i)T^{2} \) |
| 19 | \( 1 + (-1.46 - 3.29i)T + (-12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-4.11 - 8.06i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-0.0681 - 0.0495i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (0.778 - 0.208i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.526 + 5.01i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-2.24 + 5.85i)T + (-31.9 - 28.7i)T^{2} \) |
| 47 | \( 1 + (1.42 - 8.99i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.0864 + 0.133i)T + (-21.5 - 48.4i)T^{2} \) |
| 59 | \( 1 + (10.2 - 1.07i)T + (57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + 0.733iT - 61T^{2} \) |
| 67 | \( 1 + (-11.2 - 3.01i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.40 - 0.298i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-0.441 - 8.42i)T + (-72.6 + 7.63i)T^{2} \) |
| 79 | \( 1 + (-11.2 - 12.5i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.584 - 0.722i)T + (-17.2 + 81.1i)T^{2} \) |
| 89 | \( 1 + (5.38 + 16.5i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.24 - 1.65i)T + (57.0 + 78.4i)T^{2} \) |
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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
−12.78092956567205063892745528544, −11.94238235546460061253664614226, −11.04258743609612836904105680275, −9.612926935345398272888752804820, −8.476429749138284979949025431018, −7.59935623366989420266274991492, −5.81434967765360812769142928799, −5.05566411246070317076707572096, −3.86473005042662961297717726260, −1.17279877345586351013966742438,
2.61645621300489324117890286720, 4.76041709347458861356235107875, 5.14113017083375109569256840354, 6.42005784300471170757950498302, 8.013660766199508766460661590344, 9.193647929278277882901868597776, 10.34910231329024549178445598038, 11.11583725919070539359278463832, 12.06845374205191698324511004939, 13.35835455182114127959593007832
