L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.959 + 0.281i)4-s + (0.357 − 0.412i)5-s + (−3.96 + 2.54i)7-s + (0.415 + 0.909i)8-s + (−0.459 − 0.295i)10-s + (−0.698 + 4.86i)11-s + (2.05 + 1.31i)13-s + (3.08 + 3.55i)14-s + (0.841 − 0.540i)16-s + (1.58 + 0.466i)17-s + (−1.11 + 0.328i)19-s + (−0.226 + 0.496i)20-s + 4.91·22-s + (−4.79 + 0.102i)23-s + ⋯ |
L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.479 + 0.140i)4-s + (0.159 − 0.184i)5-s + (−1.49 + 0.961i)7-s + (0.146 + 0.321i)8-s + (−0.145 − 0.0933i)10-s + (−0.210 + 1.46i)11-s + (0.568 + 0.365i)13-s + (0.823 + 0.950i)14-s + (0.210 − 0.135i)16-s + (0.385 + 0.113i)17-s + (−0.256 + 0.0753i)19-s + (−0.0507 + 0.111i)20-s + 1.04·22-s + (−0.999 + 0.0213i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.658364 + 0.415965i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.658364 + 0.415965i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (4.79 - 0.102i)T \) |
good | 5 | \( 1 + (-0.357 + 0.412i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (3.96 - 2.54i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.698 - 4.86i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.05 - 1.31i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.58 - 0.466i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (1.11 - 0.328i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-2.00 - 0.588i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.66 + 5.84i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.83 - 2.11i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (1.96 - 2.26i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (4.45 - 9.76i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + (2.93 - 1.88i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (2.76 + 1.77i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (2.18 + 4.77i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.0899 - 0.625i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (0.876 + 6.09i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (10.6 - 3.13i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-3.05 - 1.96i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-3.19 - 3.68i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.310 + 0.680i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.07 + 3.54i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.51793361714342850469908198321, −10.25761951449186222969615132470, −9.607372132569289328525389792738, −9.061917304883381075132275319321, −7.82555364611835234922478654518, −6.56548697261322149489997307095, −5.65231889868258755238514955444, −4.32392360245282650281510209695, −3.11394393996434643612076072302, −1.93326906455707473892118399088,
0.51191606278133008284485566729, 3.12653140026203534558682616606, 3.98821892882497676093313309274, 5.67467646113373514838151847222, 6.31911000237816795500481727396, 7.18918296880368271864560296174, 8.283643198680707255384339807682, 9.117049702598158945789574726594, 10.34964926586135918542176737157, 10.54532506116931731774760321678