L(s) = 1 | + (−0.156 − 0.156i)2-s − 3-s − 1.95i·4-s + (2.94 − 2.94i)5-s + (0.156 + 0.156i)6-s + (3.04 − 3.04i)7-s + (−0.617 + 0.617i)8-s + 9-s − 0.920·10-s + (−2.62 + 2.02i)11-s + 1.95i·12-s + (−3.43 + 1.08i)13-s − 0.951·14-s + (−2.94 + 2.94i)15-s − 3.70·16-s + 3.38·17-s + ⋯ |
L(s) = 1 | + (−0.110 − 0.110i)2-s − 0.577·3-s − 0.975i·4-s + (1.31 − 1.31i)5-s + (0.0638 + 0.0638i)6-s + (1.14 − 1.14i)7-s + (−0.218 + 0.218i)8-s + 0.333·9-s − 0.291·10-s + (−0.791 + 0.610i)11-s + 0.563i·12-s + (−0.953 + 0.301i)13-s − 0.254·14-s + (−0.760 + 0.760i)15-s − 0.927·16-s + 0.820·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.308 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.828013 - 1.13890i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.828013 - 1.13890i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.62 - 2.02i)T \) |
| 13 | \( 1 + (3.43 - 1.08i)T \) |
good | 2 | \( 1 + (0.156 + 0.156i)T + 2iT^{2} \) |
| 5 | \( 1 + (-2.94 + 2.94i)T - 5iT^{2} \) |
| 7 | \( 1 + (-3.04 + 3.04i)T - 7iT^{2} \) |
| 17 | \( 1 - 3.38T + 17T^{2} \) |
| 19 | \( 1 + (-4.02 - 4.02i)T + 19iT^{2} \) |
| 23 | \( 1 - 4.41iT - 23T^{2} \) |
| 29 | \( 1 - 2.78iT - 29T^{2} \) |
| 31 | \( 1 + (-1.86 + 1.86i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.509 + 0.509i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.774 - 0.774i)T + 41iT^{2} \) |
| 43 | \( 1 + 9.21T + 43T^{2} \) |
| 47 | \( 1 + (-6.01 - 6.01i)T + 47iT^{2} \) |
| 53 | \( 1 - 0.0464T + 53T^{2} \) |
| 59 | \( 1 + (-4.66 - 4.66i)T + 59iT^{2} \) |
| 61 | \( 1 + 9.48iT - 61T^{2} \) |
| 67 | \( 1 + (-5.39 + 5.39i)T - 67iT^{2} \) |
| 71 | \( 1 + (6.85 - 6.85i)T - 71iT^{2} \) |
| 73 | \( 1 + (-0.740 + 0.740i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.631iT - 79T^{2} \) |
| 83 | \( 1 + (-0.524 - 0.524i)T + 83iT^{2} \) |
| 89 | \( 1 + (-0.786 - 0.786i)T + 89iT^{2} \) |
| 97 | \( 1 + (2.33 - 2.33i)T - 97iT^{2} \) |
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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
−10.64984808424238575005515978998, −9.885363763737479185238498521766, −9.606981639184959441172594263267, −8.137315080508495204668341484378, −7.17090873332624876025675108958, −5.71484983586827354785512006529, −5.16935755286329737695871571729, −4.56275878206172368513923051397, −1.85869673166045049662402731663, −1.10340019771655947098006965035,
2.32226247783971070526525735732, 2.97917160022920429311912721006, 5.04348886633437670390675132858, 5.66786481169214226700022556117, 6.80207524997589199453115003372, 7.66389375872993586683857325789, 8.647540407405085148059087840253, 9.755475730612122619338873492581, 10.57866807553589569206823961371, 11.48699753335349771388731042874