L(s) = 1 | + (1.12 − 1.12i)2-s − 3-s − 0.513i·4-s + (0.850 + 0.850i)5-s + (−1.12 + 1.12i)6-s + (0.675 + 0.675i)7-s + (1.66 + 1.66i)8-s + 9-s + 1.90·10-s + (−3.30 + 0.299i)11-s + 0.513i·12-s + (3.60 + 0.0438i)13-s + 1.51·14-s + (−0.850 − 0.850i)15-s + 4.76·16-s + 7.43·17-s + ⋯ |
L(s) = 1 | + (0.792 − 0.792i)2-s − 0.577·3-s − 0.256i·4-s + (0.380 + 0.380i)5-s + (−0.457 + 0.457i)6-s + (0.255 + 0.255i)7-s + (0.589 + 0.589i)8-s + 0.333·9-s + 0.603·10-s + (−0.995 + 0.0903i)11-s + 0.148i·12-s + (0.999 + 0.0121i)13-s + 0.404·14-s + (−0.219 − 0.219i)15-s + 1.19·16-s + 1.80·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95678 - 0.211897i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95678 - 0.211897i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 + (3.30 - 0.299i)T \) |
| 13 | \( 1 + (-3.60 - 0.0438i)T \) |
good | 2 | \( 1 + (-1.12 + 1.12i)T - 2iT^{2} \) |
| 5 | \( 1 + (-0.850 - 0.850i)T + 5iT^{2} \) |
| 7 | \( 1 + (-0.675 - 0.675i)T + 7iT^{2} \) |
| 17 | \( 1 - 7.43T + 17T^{2} \) |
| 19 | \( 1 + (2.00 - 2.00i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.77iT - 23T^{2} \) |
| 29 | \( 1 + 0.246iT - 29T^{2} \) |
| 31 | \( 1 + (7.23 + 7.23i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.872 + 0.872i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.78 - 2.78i)T - 41iT^{2} \) |
| 43 | \( 1 - 4.52T + 43T^{2} \) |
| 47 | \( 1 + (-2.70 + 2.70i)T - 47iT^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + (6.27 - 6.27i)T - 59iT^{2} \) |
| 61 | \( 1 - 3.96iT - 61T^{2} \) |
| 67 | \( 1 + (-6.55 - 6.55i)T + 67iT^{2} \) |
| 71 | \( 1 + (-2.41 - 2.41i)T + 71iT^{2} \) |
| 73 | \( 1 + (5.88 + 5.88i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.3iT - 79T^{2} \) |
| 83 | \( 1 + (2.61 - 2.61i)T - 83iT^{2} \) |
| 89 | \( 1 + (5.39 - 5.39i)T - 89iT^{2} \) |
| 97 | \( 1 + (10.7 + 10.7i)T + 97iT^{2} \) |
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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
−11.20173892497351152386448782989, −10.55987786792892686412418122845, −9.774298778987939719147277487350, −8.234800742607066746759794613575, −7.49240393213146851527479626158, −5.90046131264094511381967014873, −5.42558833335121059266818653244, −4.11955343248057237421795869172, −3.05452000486092566815758112494, −1.73962235148900244205851800222,
1.28742515391617751106861416276, 3.49293909755003342125449647246, 4.81572393252214879488797640705, 5.44746101270676312795576597485, 6.20488759439710625659027774910, 7.28083099609178824793821988409, 8.145903321163454093782687868797, 9.454212051791350795468108204919, 10.52936114080017957248019324389, 10.99706514080192150406480217350