L(s) = 1 | + (1.06 − 1.06i)2-s − 3-s − 0.261i·4-s + (0.340 + 0.340i)5-s + (−1.06 + 1.06i)6-s + (0.593 + 0.593i)7-s + (1.84 + 1.84i)8-s + 9-s + 0.724·10-s + (2.67 + 1.96i)11-s + 0.261i·12-s + (−2.47 + 2.61i)13-s + 1.26·14-s + (−0.340 − 0.340i)15-s + 4.45·16-s + 0.904·17-s + ⋯ |
L(s) = 1 | + (0.751 − 0.751i)2-s − 0.577·3-s − 0.130i·4-s + (0.152 + 0.152i)5-s + (−0.434 + 0.434i)6-s + (0.224 + 0.224i)7-s + (0.653 + 0.653i)8-s + 0.333·9-s + 0.229·10-s + (0.806 + 0.591i)11-s + 0.0755i·12-s + (−0.687 + 0.726i)13-s + 0.337·14-s + (−0.0880 − 0.0880i)15-s + 1.11·16-s + 0.219·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.88125 - 0.106421i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.88125 - 0.106421i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.67 - 1.96i)T \) |
| 13 | \( 1 + (2.47 - 2.61i)T \) |
good | 2 | \( 1 + (-1.06 + 1.06i)T - 2iT^{2} \) |
| 5 | \( 1 + (-0.340 - 0.340i)T + 5iT^{2} \) |
| 7 | \( 1 + (-0.593 - 0.593i)T + 7iT^{2} \) |
| 17 | \( 1 - 0.904T + 17T^{2} \) |
| 19 | \( 1 + (-2.48 + 2.48i)T - 19iT^{2} \) |
| 23 | \( 1 + 4.23iT - 23T^{2} \) |
| 29 | \( 1 - 3.83iT - 29T^{2} \) |
| 31 | \( 1 + (-5.60 - 5.60i)T + 31iT^{2} \) |
| 37 | \( 1 + (-5.73 + 5.73i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.91 - 2.91i)T - 41iT^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + (5.97 - 5.97i)T - 47iT^{2} \) |
| 53 | \( 1 - 3.19T + 53T^{2} \) |
| 59 | \( 1 + (5.71 - 5.71i)T - 59iT^{2} \) |
| 61 | \( 1 + 3.56iT - 61T^{2} \) |
| 67 | \( 1 + (10.0 + 10.0i)T + 67iT^{2} \) |
| 71 | \( 1 + (10.7 + 10.7i)T + 71iT^{2} \) |
| 73 | \( 1 + (-10.3 - 10.3i)T + 73iT^{2} \) |
| 79 | \( 1 + 7.59iT - 79T^{2} \) |
| 83 | \( 1 + (5.02 - 5.02i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.53 + 3.53i)T - 89iT^{2} \) |
| 97 | \( 1 + (-0.0569 - 0.0569i)T + 97iT^{2} \) |
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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
−11.45577818451067425325073867435, −10.49780497193256422812101872859, −9.635299070344851436140578045540, −8.465895821087631663891131343601, −7.23320559628268780574252383735, −6.36258030664219151732811234745, −4.93089035679323517899352525674, −4.44638049828875692456451455007, −3.01204329383290617814875961197, −1.74280598325433590533697521094,
1.22759685854150380833617168242, 3.49515449368839939404375506481, 4.66129304708847066975590041053, 5.53486119810001679752324164047, 6.21639806185811834738194993514, 7.25607048063087997277644348944, 8.084819836069008781732669783929, 9.627178711530880398899621304886, 10.16724375800250777461928275994, 11.40491018433566447132389208289