L(s) = 1 | + (−0.275 + 0.158i)3-s + (0.5 + 0.866i)5-s + 2.89·7-s + (−4.44 + 7.70i)9-s + 5.10·11-s + (−0.151 − 0.0874i)13-s + (−0.275 − 0.158i)15-s + (5.94 + 10.3i)17-s + (3.34 + 18.7i)19-s + (−0.797 + 0.460i)21-s + (−8.52 + 14.7i)23-s + (12 − 20.7i)25-s − 5.68i·27-s + (38.5 + 22.2i)29-s + 31.1i·31-s + ⋯ |
L(s) = 1 | + (−0.0917 + 0.0529i)3-s + (0.100 + 0.173i)5-s + 0.414·7-s + (−0.494 + 0.856i)9-s + 0.463·11-s + (−0.0116 − 0.00672i)13-s + (−0.0183 − 0.0105i)15-s + (0.349 + 0.606i)17-s + (0.176 + 0.984i)19-s + (−0.0379 + 0.0219i)21-s + (−0.370 + 0.641i)23-s + (0.479 − 0.831i)25-s − 0.210i·27-s + (1.32 + 0.767i)29-s + 1.00i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.30830 + 0.826148i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30830 + 0.826148i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-3.34 - 18.7i)T \) |
good | 3 | \( 1 + (0.275 - 0.158i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 - 2.89T + 49T^{2} \) |
| 11 | \( 1 - 5.10T + 121T^{2} \) |
| 13 | \( 1 + (0.151 + 0.0874i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (-5.94 - 10.3i)T + (-144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (8.52 - 14.7i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-38.5 - 22.2i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 31.1iT - 961T^{2} \) |
| 37 | \( 1 - 21.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (46.9 - 27.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-18.6 - 32.3i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-40.7 + 70.6i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-48.2 - 27.8i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-29.9 + 17.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-38.0 + 65.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (102. + 59.2i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (65.4 - 37.8i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (14.6 + 25.4i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-57.2 + 33.0i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 30.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + (8.84 + 5.10i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (128. - 74.3i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−11.70717736158990026466755542517, −10.62778784941210281251925763041, −10.04742839108288629886557474045, −8.617190867174253233123163185763, −8.014969722305128419791670282199, −6.74901431973253637646504825098, −5.65995738231737544707615930747, −4.62459706923003704702182167972, −3.18783036029967824064938705486, −1.62418567435985871527786141012,
0.819888945744019031560038321134, 2.68350368831210221323304851382, 4.11590298658617140626371755730, 5.32611783762862693041791771433, 6.40208390160890462718742213172, 7.40648978018492637906392433871, 8.663642099316022112081176378606, 9.300775805039339410452455632263, 10.41714853220462668615289554696, 11.57809307881668243481781034532