| L(s) = 1 | − 0.816i·2-s + 1.53i·3-s + 1.33·4-s + 1.25·6-s − 5.03i·7-s − 2.72i·8-s + 0.633·9-s − 3.03·11-s + 2.05i·12-s − 4.57i·13-s − 4.11·14-s + 0.443·16-s + 1.07i·17-s − 0.517i·18-s − 19-s + ⋯ |
| L(s) = 1 | − 0.577i·2-s + 0.888i·3-s + 0.666·4-s + 0.512·6-s − 1.90i·7-s − 0.962i·8-s + 0.211·9-s − 0.914·11-s + 0.592i·12-s − 1.26i·13-s − 1.09·14-s + 0.110·16-s + 0.261i·17-s − 0.121i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40537 - 0.868570i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40537 - 0.868570i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.816iT - 2T^{2} \) |
| 3 | \( 1 - 1.53iT - 3T^{2} \) |
| 7 | \( 1 + 5.03iT - 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 + 4.57iT - 13T^{2} \) |
| 17 | \( 1 - 1.07iT - 17T^{2} \) |
| 23 | \( 1 - 4.11iT - 23T^{2} \) |
| 29 | \( 1 - 1.07T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 + 0.0947iT - 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 5.03iT - 43T^{2} \) |
| 47 | \( 1 - 12.2iT - 47T^{2} \) |
| 53 | \( 1 + 4.09iT - 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 + 5.69T + 61T^{2} \) |
| 67 | \( 1 + 5.28iT - 67T^{2} \) |
| 71 | \( 1 + 5.67T + 71T^{2} \) |
| 73 | \( 1 - 9.07iT - 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 - 1.95iT - 83T^{2} \) |
| 89 | \( 1 - 2.18T + 89T^{2} \) |
| 97 | \( 1 - 2.16iT - 97T^{2} \) |
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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
−10.68508678052646327914725947634, −10.24176519686894333784507045873, −9.636751698930422513120718075475, −7.85487925663989610498647796697, −7.42306879260007566897751057013, −6.18643545151655738158417793971, −4.77859203708049122191010818547, −3.86138305339665772217166338092, −2.96093173373823314272777587970, −1.06652075441511720032289296724,
2.03021432719410596123109253626, 2.60797115770489139217578981990, 4.79807025200257499845734219525, 5.89108879204788481076558829456, 6.50667269227113230389616076624, 7.41192332638514318486043965642, 8.320534659911191967003200599899, 9.029211097868177684074537713507, 10.31165477903737496879319929333, 11.50373142979946216823457243994
