L(s) = 1 | + (1.70 + 2.94i)3-s + (2.15 − 3.73i)5-s + (1.43 − 6.85i)7-s + (−1.28 + 2.23i)9-s + (15.4 − 8.94i)11-s + 3.25·13-s + 14.6·15-s + (−13.6 + 7.86i)17-s + (−0.778 + 1.34i)19-s + (22.6 − 7.43i)21-s + (−20.7 + 35.8i)23-s + (3.18 + 5.50i)25-s + 21.8·27-s + 3.74i·29-s + (0.0145 − 0.00838i)31-s + ⋯ |
L(s) = 1 | + (0.567 + 0.982i)3-s + (0.431 − 0.747i)5-s + (0.204 − 0.978i)7-s + (−0.143 + 0.248i)9-s + (1.40 − 0.813i)11-s + 0.250·13-s + 0.979·15-s + (−0.801 + 0.462i)17-s + (−0.0409 + 0.0709i)19-s + (1.07 − 0.354i)21-s + (−0.900 + 1.55i)23-s + (0.127 + 0.220i)25-s + 0.809·27-s + 0.129i·29-s + (0.000468 − 0.000270i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0168i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.11830 - 0.0177990i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.11830 - 0.0177990i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.43 + 6.85i)T \) |
good | 3 | \( 1 + (-1.70 - 2.94i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-2.15 + 3.73i)T + (-12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-15.4 + 8.94i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 3.25T + 169T^{2} \) |
| 17 | \( 1 + (13.6 - 7.86i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (0.778 - 1.34i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (20.7 - 35.8i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 3.74iT - 841T^{2} \) |
| 31 | \( 1 + (-0.0145 + 0.00838i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (1.16 + 0.674i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 70.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 13.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (30.9 + 17.8i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-39.7 + 22.9i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-34.3 - 59.4i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (48.0 - 83.2i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (12.0 - 6.97i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 75.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (46.0 - 26.5i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (11.6 - 20.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 102.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (76.6 + 44.2i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 140. iT - 9.40e3T^{2} \) |
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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
−11.89770015110680036194107931039, −10.91317913859502215171082949211, −9.920548821412674957090005253668, −9.104657260478273864967838706519, −8.462336961158835991483889926044, −6.96152765876726820718798561440, −5.65264269464528784419171375197, −4.22309359471935586998795689924, −3.64682169333829377915720011182, −1.35557436274588786896393274427,
1.80634318410256910302443363107, 2.69483709667931724450900695614, 4.54007158457306252829906101620, 6.34894549459050887585974915374, 6.75145250054567586042997618791, 8.075513410721802435388958194297, 8.954092050041647999186859151163, 9.960747736207166867747188680346, 11.24976548934090565820868745422, 12.18067083623868668334660067335