L(s) = 1 | + 2-s + (−1.29 − 1.14i)3-s + 4-s + (1.07 − 1.96i)5-s + (−1.29 − 1.14i)6-s + 8-s + (0.375 + 2.97i)9-s + (1.07 − 1.96i)10-s + 0.115i·11-s + (−1.29 − 1.14i)12-s + 5.60·13-s + (−3.64 + 1.31i)15-s + 16-s − 1.39i·17-s + (0.375 + 2.97i)18-s − 1.52i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.750 − 0.661i)3-s + 0.5·4-s + (0.479 − 0.877i)5-s + (−0.530 − 0.467i)6-s + 0.353·8-s + (0.125 + 0.992i)9-s + (0.339 − 0.620i)10-s + 0.0348i·11-s + (−0.375 − 0.330i)12-s + 1.55·13-s + (−0.940 + 0.340i)15-s + 0.250·16-s − 0.337i·17-s + (0.0884 + 0.701i)18-s − 0.349i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.398867241\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.398867241\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.29 + 1.14i)T \) |
| 5 | \( 1 + (-1.07 + 1.96i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 0.115iT - 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 + 1.39iT - 17T^{2} \) |
| 19 | \( 1 + 1.52iT - 19T^{2} \) |
| 23 | \( 1 - 6.58T + 23T^{2} \) |
| 29 | \( 1 - 8.97iT - 29T^{2} \) |
| 31 | \( 1 + 7.15iT - 31T^{2} \) |
| 37 | \( 1 - 1.70iT - 37T^{2} \) |
| 41 | \( 1 + 3.82T + 41T^{2} \) |
| 43 | \( 1 + 12.1iT - 43T^{2} \) |
| 47 | \( 1 - 2.37iT - 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 3.32T + 59T^{2} \) |
| 61 | \( 1 - 5.12iT - 61T^{2} \) |
| 67 | \( 1 + 0.689iT - 67T^{2} \) |
| 71 | \( 1 + 6.77iT - 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 6.24iT - 83T^{2} \) |
| 89 | \( 1 + 6.83T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−9.178450327144766579122369201350, −8.540034302893608414673993386104, −7.50774374822296191186613026843, −6.65654600912952319383448165553, −5.95028662885815488840259991247, −5.22738499041217946286564155463, −4.55337568003464628212759128921, −3.28484416380508373397669892776, −1.88957733523596298145504718883, −0.948202739767666964225165804326,
1.42514355161826600431884792428, 3.01064433277364389660455829044, 3.68453452268906557690702739403, 4.68048327165738248649082248445, 5.64784879147690410501351917895, 6.28590749429401374823852340443, 6.77409075982574148245540531247, 7.991847790422367576890384804047, 9.072743029019136096516060425467, 9.918895585308575800668097429525