L(s) = 1 | + (1.40 + 0.146i)2-s + (−0.169 + 1.72i)3-s + (1.95 + 0.413i)4-s + 2.76i·5-s + (−0.492 + 2.39i)6-s − 3.82i·7-s + (2.69 + 0.868i)8-s + (−2.94 − 0.585i)9-s + (−0.406 + 3.89i)10-s + 2.92·11-s + (−1.04 + 3.30i)12-s − 4.70·13-s + (0.561 − 5.38i)14-s + (−4.76 − 0.469i)15-s + (3.65 + 1.61i)16-s + 4.06i·17-s + ⋯ |
L(s) = 1 | + (0.994 + 0.103i)2-s + (−0.0980 + 0.995i)3-s + (0.978 + 0.206i)4-s + 1.23i·5-s + (−0.200 + 0.979i)6-s − 1.44i·7-s + (0.951 + 0.307i)8-s + (−0.980 − 0.195i)9-s + (−0.128 + 1.23i)10-s + 0.883·11-s + (−0.301 + 0.953i)12-s − 1.30·13-s + (0.150 − 1.43i)14-s + (−1.23 − 0.121i)15-s + (0.914 + 0.404i)16-s + 0.985i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75839 + 1.28816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75839 + 1.28816i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.146i)T \) |
| 3 | \( 1 + (0.169 - 1.72i)T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 2.76iT - 5T^{2} \) |
| 7 | \( 1 + 3.82iT - 7T^{2} \) |
| 11 | \( 1 - 2.92T + 11T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 - 4.06iT - 17T^{2} \) |
| 19 | \( 1 + 3.75iT - 19T^{2} \) |
| 29 | \( 1 + 8.01iT - 29T^{2} \) |
| 31 | \( 1 + 1.15iT - 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 1.54iT - 41T^{2} \) |
| 43 | \( 1 + 2.22iT - 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 6.30iT - 53T^{2} \) |
| 59 | \( 1 - 7.03T + 59T^{2} \) |
| 61 | \( 1 + 6.15T + 61T^{2} \) |
| 67 | \( 1 - 1.65iT - 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 5.68T + 73T^{2} \) |
| 79 | \( 1 - 3.06iT - 79T^{2} \) |
| 83 | \( 1 - 2.88T + 83T^{2} \) |
| 89 | \( 1 - 8.17iT - 89T^{2} \) |
| 97 | \( 1 + 4.28T + 97T^{2} \) |
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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
−11.82840054779655175472894366010, −11.12885043826742219456958091424, −10.40063950168248422690798464766, −9.699560681798489730963646866523, −7.84085211148508938402323642372, −6.91597945440058640371107635348, −6.09509575573344560092422951165, −4.53367474090575749372038293887, −3.89640699352924606759486140830, −2.71515072502769208720403985724,
1.59760686949204528089651730469, 2.85723375805134033820646905272, 4.78125889285171791312770380749, 5.48982680265511720967086565042, 6.48318437574042705858523256030, 7.65246610579502025488933267684, 8.732359399049489295124413219781, 9.663699167718258004285044302390, 11.50456011514454360369190680212, 11.99576690340289133102641610459