L(s) = 1 | + (−1.41 + 5i)3-s + (−23 − 14.1i)9-s − 18i·11-s − 107. i·17-s + 127.·19-s − 125·25-s + (103. − 95i)27-s + (90 + 25.4i)33-s + 56.5i·41-s + 483.·43-s + 343·49-s + (537. + 152i)51-s + (−180. + 636. i)57-s − 846i·59-s + 1.09e3·67-s + ⋯ |
L(s) = 1 | + (−0.272 + 0.962i)3-s + (−0.851 − 0.523i)9-s − 0.493i·11-s − 1.53i·17-s + 1.53·19-s − 25-s + (0.735 − 0.677i)27-s + (0.474 + 0.134i)33-s + 0.215i·41-s + 1.71·43-s + 49-s + (1.47 + 0.417i)51-s + (−0.418 + 1.47i)57-s − 1.86i·59-s + 1.99·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.457995901\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.457995901\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 - 5i)T \) |
good | 5 | \( 1 + 125T^{2} \) |
| 7 | \( 1 - 343T^{2} \) |
| 11 | \( 1 + 18iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 + 107. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 - 56.5iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 483.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 846iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 1.09e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 430T + 3.89e5T^{2} \) |
| 79 | \( 1 - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.35e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.32e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.91e3T + 9.12e5T^{2} \) |
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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
−11.01155328399017589949582413090, −9.729116762451639454190587020097, −9.412942788330065927245775002492, −8.190439836710966278987596750997, −7.09107231736497162640078807037, −5.77824610563574928417970298036, −5.06102458415549858583442647704, −3.84021682710577073892891907720, −2.77506563244789333450532936977, −0.61095474250302536327077017672,
1.12219058602106780854641354648, 2.35459173960219017974957033223, 3.87032060020497671754576868261, 5.36909914044477821358717535262, 6.16634286078935731291759537324, 7.28865293552701875292825605761, 7.933727857767893288667867694332, 9.021307968498768461407663544507, 10.13479304597531054393216565220, 11.08731700369923527498882510465