L(s) = 1 | + (−0.599 − 1.28i)2-s + (−0.468 + 1.66i)3-s + (−1.28 + 1.53i)4-s + (2.41 − 0.400i)6-s + 0.936i·7-s + (2.73 + 0.719i)8-s + (−2.56 − 1.56i)9-s − 4.27·11-s + (−1.96 − 2.85i)12-s − 3.12·13-s + (1.19 − 0.561i)14-s + (−0.719 − 3.93i)16-s + 2i·17-s + (−0.463 + 4.21i)18-s + 4.27i·19-s + ⋯ |
L(s) = 1 | + (−0.424 − 0.905i)2-s + (−0.270 + 0.962i)3-s + (−0.640 + 0.768i)4-s + (0.986 − 0.163i)6-s + 0.353i·7-s + (0.967 + 0.254i)8-s + (−0.853 − 0.520i)9-s − 1.28·11-s + (−0.566 − 0.824i)12-s − 0.866·13-s + (0.320 − 0.150i)14-s + (−0.179 − 0.983i)16-s + 0.485i·17-s + (−0.109 + 0.994i)18-s + 0.979i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.173307 + 0.329374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.173307 + 0.329374i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.599 + 1.28i)T \) |
| 3 | \( 1 + (0.468 - 1.66i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.936iT - 7T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4.27iT - 19T^{2} \) |
| 23 | \( 1 + 7.60T + 23T^{2} \) |
| 29 | \( 1 - 5.12iT - 29T^{2} \) |
| 31 | \( 1 - 2.39iT - 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 7.12iT - 41T^{2} \) |
| 43 | \( 1 - 1.46iT - 43T^{2} \) |
| 47 | \( 1 - 0.936T + 47T^{2} \) |
| 53 | \( 1 - 4.24iT - 53T^{2} \) |
| 59 | \( 1 + 7.19T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 - 5.20iT - 67T^{2} \) |
| 71 | \( 1 - 6.67T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 9.06iT - 79T^{2} \) |
| 83 | \( 1 - 4.68T + 83T^{2} \) |
| 89 | \( 1 - 6.24iT - 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−12.09742652096447331577419305170, −10.80905031174912895067488697493, −10.32515300610893261235417179891, −9.548403868939200975405928098983, −8.511929255288157834020332559266, −7.65014392703888512268570491526, −5.78525142880518885569640046816, −4.79841076686813089934800056249, −3.62676430646168778544878298472, −2.36519375137095092997258297346,
0.30608622156097079726339614810, 2.36764872704517405538761170340, 4.66104511109969010822380245253, 5.64651560968772193466329559823, 6.66539556658376431598065448425, 7.66119027476579965686524176123, 8.049309674512852047592161169015, 9.442136116203789078910832880450, 10.35903322841464958423255742732, 11.37078416809400436548166559455