| L(s) = 1 | + 1.23i·2-s − 0.363i·3-s + 0.477·4-s + (1.29 − 1.82i)5-s + 0.449·6-s + 2.58i·7-s + 3.05i·8-s + 2.86·9-s + (2.24 + 1.59i)10-s − 0.173i·12-s − 2.75i·13-s − 3.19·14-s + (−0.663 − 0.471i)15-s − 2.81·16-s − 3.85i·17-s + 3.53i·18-s + ⋯ |
| L(s) = 1 | + 0.872i·2-s − 0.210i·3-s + 0.238·4-s + (0.579 − 0.815i)5-s + 0.183·6-s + 0.977i·7-s + 1.08i·8-s + 0.955·9-s + (0.711 + 0.505i)10-s − 0.0501i·12-s − 0.765i·13-s − 0.852·14-s + (−0.171 − 0.121i)15-s − 0.704·16-s − 0.934i·17-s + 0.834i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.579 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.579 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75929 + 0.908251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75929 + 0.908251i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.29 + 1.82i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.23iT - 2T^{2} \) |
| 3 | \( 1 + 0.363iT - 3T^{2} \) |
| 7 | \( 1 - 2.58iT - 7T^{2} \) |
| 13 | \( 1 + 2.75iT - 13T^{2} \) |
| 17 | \( 1 + 3.85iT - 17T^{2} \) |
| 19 | \( 1 + 0.277T + 19T^{2} \) |
| 23 | \( 1 - 8.40iT - 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 - 0.564T + 31T^{2} \) |
| 37 | \( 1 + 0.522iT - 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 - 2.54iT - 43T^{2} \) |
| 47 | \( 1 + 4.92iT - 47T^{2} \) |
| 53 | \( 1 - 8.72iT - 53T^{2} \) |
| 59 | \( 1 + 7.50T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 3.20iT - 67T^{2} \) |
| 71 | \( 1 + 8.40T + 71T^{2} \) |
| 73 | \( 1 + 13.0iT - 73T^{2} \) |
| 79 | \( 1 - 9.70T + 79T^{2} \) |
| 83 | \( 1 - 3.29iT - 83T^{2} \) |
| 89 | \( 1 + 2.48T + 89T^{2} \) |
| 97 | \( 1 + 10.9iT - 97T^{2} \) |
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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
−10.73034185000668967159947146029, −9.627663780872581032050913876812, −8.982053339805655222520608116767, −7.947683818857726072480978468705, −7.29878717464438344806245278014, −6.13952641215094186423154477268, −5.52643720358721495363180687981, −4.67537637020268704296163825216, −2.79581971450744370846750751402, −1.55776196138830821402920479356,
1.38396351227975638019251832274, 2.50556973495352807517076776072, 3.75265556654776585212126995640, 4.48928111412439945334710620241, 6.33460552204897236325639893680, 6.77207202730745775955014629678, 7.70485158832549623611843094952, 9.183947160380146649190058151332, 10.09015763859017768903834653167, 10.54496654383599855120009082463
