L(s) = 1 | + (−2.21 − 1.27i)2-s + (2.59 − 1.5i)3-s + (−0.732 − 1.26i)4-s + (5.74 + 9.58i)5-s − 7.66·6-s + (−2.98 + 18.2i)7-s + 24.1i·8-s + (4.5 − 7.79i)9-s + (−0.470 − 28.5i)10-s + (0.664 + 1.15i)11-s + (−3.80 − 2.19i)12-s + 46.0i·13-s + (29.9 − 36.6i)14-s + (29.3 + 16.2i)15-s + (25.0 − 43.4i)16-s + (77.3 − 44.6i)17-s + ⋯ |
L(s) = 1 | + (−0.782 − 0.451i)2-s + (0.499 − 0.288i)3-s + (−0.0916 − 0.158i)4-s + (0.514 + 0.857i)5-s − 0.521·6-s + (−0.161 + 0.986i)7-s + 1.06i·8-s + (0.166 − 0.288i)9-s + (−0.0148 − 0.903i)10-s + (0.0182 + 0.0315i)11-s + (−0.0916 − 0.0528i)12-s + 0.982i·13-s + (0.572 − 0.699i)14-s + (0.504 + 0.280i)15-s + (0.391 − 0.678i)16-s + (1.10 − 0.636i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.14323 + 0.256492i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14323 + 0.256492i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.59 + 1.5i)T \) |
| 5 | \( 1 + (-5.74 - 9.58i)T \) |
| 7 | \( 1 + (2.98 - 18.2i)T \) |
good | 2 | \( 1 + (2.21 + 1.27i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.664 - 1.15i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 46.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-77.3 + 44.6i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (27.1 - 47.1i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-132. - 76.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (57.7 + 100. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (112. + 64.8i)T + (2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 128.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 428. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (487. + 281. i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (77.5 - 44.7i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-292. - 506. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-170. + 296. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (83.8 - 48.4i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 673.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-494. + 285. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-298. + 517. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 801. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-69.1 + 119. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.52e3iT - 9.12e5T^{2} \) |
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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
−13.50646127473078090761379398798, −12.05809708920615496801898805548, −11.13367397443698309572623182263, −9.828412724586365437453214716220, −9.314240366541570475054995319556, −8.133451167606373536572814066186, −6.69153140131521698487399999796, −5.38037680526209855368335547841, −2.98037559997402151357042378879, −1.73127777722281226382552100527,
0.871880455523763576858923390757, 3.46341928532526394920397606667, 4.93426511650112413130988831445, 6.73528607492390099014670279597, 8.002510434511980497441962673663, 8.704162947079418802166165984492, 9.839893131179268772763966213605, 10.51913645511510627173699895290, 12.56501567105591873604601630284, 13.13269291641660480136839481056