L(s) = 1 | + 2.36i·2-s − i·3-s − 3.58·4-s + (0.979 + 2.01i)5-s + 2.36·6-s − 1.71i·7-s − 3.75i·8-s − 9-s + (−4.75 + 2.31i)10-s + 3.58i·12-s + 3.53i·13-s + 4.05·14-s + (2.01 − 0.979i)15-s + 1.69·16-s − 0.373i·17-s − 2.36i·18-s + ⋯ |
L(s) = 1 | + 1.67i·2-s − 0.577i·3-s − 1.79·4-s + (0.437 + 0.899i)5-s + 0.965·6-s − 0.647i·7-s − 1.32i·8-s − 0.333·9-s + (−1.50 + 0.731i)10-s + 1.03i·12-s + 0.979i·13-s + 1.08·14-s + (0.519 − 0.252i)15-s + 0.423·16-s − 0.0905i·17-s − 0.557i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.219997248\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.219997248\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (-0.979 - 2.01i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.36iT - 2T^{2} \) |
| 7 | \( 1 + 1.71iT - 7T^{2} \) |
| 13 | \( 1 - 3.53iT - 13T^{2} \) |
| 17 | \( 1 + 0.373iT - 17T^{2} \) |
| 19 | \( 1 - 5.76T + 19T^{2} \) |
| 23 | \( 1 - 3.80iT - 23T^{2} \) |
| 29 | \( 1 - 3.51T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 11.5iT - 37T^{2} \) |
| 41 | \( 1 + 2.41T + 41T^{2} \) |
| 43 | \( 1 - 0.224iT - 43T^{2} \) |
| 47 | \( 1 + 4.67iT - 47T^{2} \) |
| 53 | \( 1 - 2.44iT - 53T^{2} \) |
| 59 | \( 1 + 5.35T + 59T^{2} \) |
| 61 | \( 1 + 3.40T + 61T^{2} \) |
| 67 | \( 1 - 9.08iT - 67T^{2} \) |
| 71 | \( 1 + 8.77T + 71T^{2} \) |
| 73 | \( 1 - 14.9iT - 73T^{2} \) |
| 79 | \( 1 + 8.79T + 79T^{2} \) |
| 83 | \( 1 + 7.44iT - 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 14.6iT - 97T^{2} \) |
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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
−9.480444679200944887252015272333, −8.701755443191246079546903288672, −7.73301102098775467736891402577, −7.10291778276570598089427625879, −6.86203173085318959591858416502, −5.91004550123203039575748548265, −5.27738572337343416466123710330, −4.14926304046020640489973376762, −3.04138499627512827249355443571, −1.53245711595591034883648480437,
0.46044310752619831436508058878, 1.71062955913968495474442543892, 2.71741057002202776069242638785, 3.53969176306688702899628767405, 4.52490075455489046548111080841, 5.28871049240416640237748783994, 5.89678174745978539029600887547, 7.53661257538159542657375349781, 8.601434544000769735513367093910, 9.111975463771339539995817029784