| L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (−0.5 + 0.866i)5-s − 2.89·7-s − 2.82i·8-s + (1.22 − 0.707i)10-s + 5.10·11-s + (−0.151 + 0.0874i)13-s + (3.55 + 2.04i)14-s + (−2.00 + 3.46i)16-s + (−5.94 + 10.3i)17-s + (−3.34 + 18.7i)19-s − 1.99·20-s + (−6.24 − 3.60i)22-s + (−8.52 − 14.7i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.100 + 0.173i)5-s − 0.414·7-s − 0.353i·8-s + (0.122 − 0.0707i)10-s + 0.463·11-s + (−0.0116 + 0.00672i)13-s + (0.253 + 0.146i)14-s + (−0.125 + 0.216i)16-s + (−0.349 + 0.606i)17-s + (−0.176 + 0.984i)19-s − 0.0999·20-s + (−0.283 − 0.163i)22-s + (−0.370 − 0.641i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0791 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0791 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.562280 + 0.519387i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.562280 + 0.519387i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.34 - 18.7i)T \) |
| good | 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
−11.27776429507522000269970476711, −10.65812037181999362350884030096, −9.643342821712821757115435220375, −8.876038857965984073141212582092, −7.86017271116344144506169798465, −6.85150872107706377793666820379, −5.85820283901217907867044481787, −4.20894740510293469969930542243, −3.10057278563229079827014227284, −1.54409086668494526450394552213,
0.43755326700475913569260360500, 2.30264369545649628690076599498, 3.94909407165228189820233586554, 5.29070170011977634521837289591, 6.43813854753557106045798028210, 7.25976684050067267430286823567, 8.331425718206933538621419719137, 9.267501126591975729492223358910, 9.887702777501840254648486912452, 11.10238620506190106250171130667
