L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (0.866 + 0.5i)5-s + (−0.866 + 0.5i)8-s + (−0.342 + 0.939i)9-s + (−0.939 − 0.342i)10-s + (0.592 + 1.62i)13-s + (0.766 − 0.642i)16-s + (−1.43 − 0.524i)17-s + (0.173 − 0.984i)18-s + (0.984 + 0.173i)20-s + (0.499 + 0.866i)25-s + (−0.866 − 1.5i)26-s + (0.469 − 1.75i)29-s + (−0.642 + 0.766i)32-s + ⋯ |
L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (0.866 + 0.5i)5-s + (−0.866 + 0.5i)8-s + (−0.342 + 0.939i)9-s + (−0.939 − 0.342i)10-s + (0.592 + 1.62i)13-s + (0.766 − 0.642i)16-s + (−1.43 − 0.524i)17-s + (0.173 − 0.984i)18-s + (0.984 + 0.173i)20-s + (0.499 + 0.866i)25-s + (−0.866 − 1.5i)26-s + (0.469 − 1.75i)29-s + (−0.642 + 0.766i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6974916599\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6974916599\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 37 | \( 1 + (0.342 + 0.939i)T \) |
good | 3 | \( 1 + (0.342 - 0.939i)T^{2} \) |
| 7 | \( 1 + (-0.984 + 0.173i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 19 | \( 1 + (0.342 - 0.939i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.469 + 1.75i)T + (-0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.439 - 1.20i)T + (-0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.515 - 0.0451i)T + (0.984 + 0.173i)T^{2} \) |
| 59 | \( 1 + (0.984 + 0.173i)T^{2} \) |
| 61 | \( 1 + (0.357 - 0.766i)T + (-0.642 - 0.766i)T^{2} \) |
| 67 | \( 1 + (0.984 - 0.173i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 79 | \( 1 + (-0.984 + 0.173i)T^{2} \) |
| 83 | \( 1 + (0.642 - 0.766i)T^{2} \) |
| 89 | \( 1 + (-0.173 - 0.0151i)T + (0.984 + 0.173i)T^{2} \) |
| 97 | \( 1 + (-0.342 + 0.592i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.76067444941930635427768490880, −9.692845659836932800759351100759, −9.118629811949836707827520449992, −8.305353373127105000772759056998, −7.19501785084613290115878201225, −6.50980164620071267723099364071, −5.73044571392788577620396598360, −4.41767307695046762933185835368, −2.58921980027080127982068932535, −1.89788314147824459590205165907,
1.08779499773764868223892473897, 2.52320938570822896644558510632, 3.64144473583051867047408231064, 5.31024753210860842150392367936, 6.19132483606818077106362419926, 6.91318132309444430171713138528, 8.279667606751972608772341154993, 8.760247007851554394987827014167, 9.455686512221119356226464062486, 10.49781458485541886015581776735