| L(s) = 1 | − 1.65i·2-s − 2.37i·3-s − 0.726·4-s − 3.92·6-s − 0.377i·7-s − 2.10i·8-s − 2.65·9-s − 1.37·11-s + 1.72i·12-s − 2.82i·13-s − 0.622·14-s − 4.92·16-s + 6.37i·17-s + 4.37i·18-s − 19-s + ⋯ |
| L(s) = 1 | − 1.16i·2-s − 1.37i·3-s − 0.363·4-s − 1.60·6-s − 0.142i·7-s − 0.743i·8-s − 0.883·9-s − 0.415·11-s + 0.498i·12-s − 0.782i·13-s − 0.166·14-s − 1.23·16-s + 1.54i·17-s + 1.03i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.312551 + 1.32399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.312551 + 1.32399i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.65iT - 2T^{2} \) |
| 3 | \( 1 + 2.37iT - 3T^{2} \) |
| 7 | \( 1 + 0.377iT - 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 6.37iT - 17T^{2} \) |
| 23 | \( 1 + 6.19iT - 23T^{2} \) |
| 29 | \( 1 - 3.37T + 29T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 - 5.58iT - 37T^{2} \) |
| 41 | \( 1 - 8.50T + 41T^{2} \) |
| 43 | \( 1 - 12.1iT - 43T^{2} \) |
| 47 | \( 1 + 6.87iT - 47T^{2} \) |
| 53 | \( 1 + 11.5iT - 53T^{2} \) |
| 59 | \( 1 + 6.05T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 3.22iT - 67T^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 - 3.19iT - 73T^{2} \) |
| 79 | \( 1 - 6.71T + 79T^{2} \) |
| 83 | \( 1 + 18.2iT - 83T^{2} \) |
| 89 | \( 1 + 1.50T + 89T^{2} \) |
| 97 | \( 1 + 11.7iT - 97T^{2} \) |
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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
−10.56538979430942364465327203252, −10.04860248754623878603139696072, −8.542931506594364414974769910122, −7.86792837818423058749137653550, −6.74951786250809037048135419179, −6.05700799153281735036208714066, −4.38614780625759664393458094098, −3.01046236395416844157784638829, −2.03883293802583197803124029126, −0.843513459550189060569675064243,
2.63238659433193840400727091627, 4.12339578204318141889404929651, 5.03770085644675788851732631514, 5.75573087197535777027025033939, 6.98463263574236400700861476918, 7.75567222076213047027908721680, 9.038614799439906842741606947416, 9.384147476805564599127930828944, 10.58099782907346454090115166840, 11.29555014677689249216835731382
