| L(s) = 1 | + (−0.0603 + 0.574i)2-s + (0.641 + 1.60i)3-s + (1.62 + 0.346i)4-s + (−0.314 − 2.98i)5-s + (−0.963 + 0.271i)6-s + (−0.779 + 0.865i)7-s + (−0.654 + 2.01i)8-s + (−2.17 + 2.06i)9-s + 1.73·10-s + (−0.565 − 3.26i)11-s + (0.487 + 2.84i)12-s + (−3.60 + 1.60i)13-s + (−0.450 − 0.499i)14-s + (4.60 − 2.42i)15-s + (1.92 + 0.857i)16-s + (0.254 − 0.184i)17-s + ⋯ |
| L(s) = 1 | + (−0.0427 + 0.406i)2-s + (0.370 + 0.928i)3-s + (0.814 + 0.173i)4-s + (−0.140 − 1.33i)5-s + (−0.393 + 0.110i)6-s + (−0.294 + 0.327i)7-s + (−0.231 + 0.712i)8-s + (−0.726 + 0.687i)9-s + 0.548·10-s + (−0.170 − 0.985i)11-s + (0.140 + 0.821i)12-s + (−0.999 + 0.445i)13-s + (−0.120 − 0.133i)14-s + (1.18 − 0.625i)15-s + (0.481 + 0.214i)16-s + (0.0616 − 0.0447i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04859 + 0.518547i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04859 + 0.518547i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.641 - 1.60i)T \) |
| 11 | \( 1 + (0.565 + 3.26i)T \) |
| good | 2 | \( 1 + (0.0603 - 0.574i)T + (-1.95 - 0.415i)T^{2} \) |
| 5 | \( 1 + (0.314 + 2.98i)T + (-4.89 + 1.03i)T^{2} \) |
| 7 | \( 1 + (0.779 - 0.865i)T + (-0.731 - 6.96i)T^{2} \) |
| 13 | \( 1 + (3.60 - 1.60i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-0.254 + 0.184i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.96 + 6.05i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.0427 + 0.0740i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.15 - 5.72i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (-5.96 + 2.65i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (1.92 + 5.93i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.88 - 4.31i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-3.39 - 5.87i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.320 + 0.0681i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-1.95 - 1.42i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.24 - 0.477i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (12.5 + 5.56i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (5.83 - 10.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.05 - 5.12i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.910 - 2.80i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.928 + 8.83i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-4.52 - 2.01i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 - 2.12T + 89T^{2} \) |
| 97 | \( 1 + (-0.00928 + 0.0883i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−14.28567395893325739619907563505, −13.01543842697196226453862371951, −11.85987285191697867756080160068, −10.94755015797345865623964932883, −9.393914203797880424901939636582, −8.711181747485364364560481059857, −7.53911744395666783972156061683, −5.77924677951670112747475387672, −4.69486529562146101372067545008, −2.84376586585030215448033154707,
2.19396249900212734961064907901, 3.33763624208864984602607677907, 6.08268229290146858478780619534, 7.16209097092196172350401077690, 7.65847471717181150620286495406, 9.825470176581278325745937096001, 10.50609906408579833388174829450, 11.83930815492243944190224334258, 12.41707398983570047486779898246, 13.77881619139159869996451098198
