L(s) = 1 | + (0.770 + 1.84i)2-s + (1.12 − 2.78i)3-s + (−2.81 + 2.84i)4-s + (5.99 − 0.0674i)6-s + (4.75 − 4.75i)7-s + (−7.41 − 2.99i)8-s + (−6.46 − 6.25i)9-s + 11.9·11-s + (4.74 + 11.0i)12-s + (4.22 − 4.22i)13-s + (12.4 + 5.10i)14-s + (−0.188 − 15.9i)16-s + (9.35 − 9.35i)17-s + (6.56 − 16.7i)18-s − 1.48·19-s + ⋯ |
L(s) = 1 | + (0.385 + 0.922i)2-s + (0.375 − 0.927i)3-s + (−0.702 + 0.711i)4-s + (0.999 − 0.0112i)6-s + (0.678 − 0.678i)7-s + (−0.927 − 0.374i)8-s + (−0.718 − 0.695i)9-s + 1.08·11-s + (0.395 + 0.918i)12-s + (0.324 − 0.324i)13-s + (0.887 + 0.364i)14-s + (−0.0117 − 0.999i)16-s + (0.550 − 0.550i)17-s + (0.364 − 0.931i)18-s − 0.0783·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.21270 - 0.162571i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.21270 - 0.162571i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.770 - 1.84i)T \) |
| 3 | \( 1 + (-1.12 + 2.78i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-4.75 + 4.75i)T - 49iT^{2} \) |
| 11 | \( 1 - 11.9T + 121T^{2} \) |
| 13 | \( 1 + (-4.22 + 4.22i)T - 169iT^{2} \) |
| 17 | \( 1 + (-9.35 + 9.35i)T - 289iT^{2} \) |
| 19 | \( 1 + 1.48T + 361T^{2} \) |
| 23 | \( 1 + (-11.6 + 11.6i)T - 529iT^{2} \) |
| 29 | \( 1 - 39.3T + 841T^{2} \) |
| 31 | \( 1 - 43.6iT - 961T^{2} \) |
| 37 | \( 1 + (49.1 + 49.1i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 27.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-8.84 - 8.84i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-15.0 - 15.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-14.6 - 14.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 61.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 84.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + (65.7 - 65.7i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 14.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (16.0 - 16.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 9.32T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-12.7 + 12.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 52.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + (6.90 + 6.90i)T + 9.40e3iT^{2} \) |
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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
−11.95692058626022981936811257148, −10.63074383972789152476931207744, −9.089430292490621155929024341567, −8.464545928787320909659578737761, −7.39436638045036609041462763792, −6.83494547950961809400496227517, −5.70782323407801346920248488876, −4.38989365603402157004809827569, −3.14581045237010256702591343384, −1.07560287616471204337930401480,
1.69094998400915713942457683966, 3.13958776238839648536585046533, 4.19609780682246314869728181184, 5.13254811644380207949889495133, 6.22903945952362621572087348477, 8.220492972869011049294995745968, 8.942725025987629198135599546002, 9.746307924579466652336649935170, 10.67993635624377477931083055144, 11.59165682002485744217406477471