| L(s) = 1 | − 1.14·2-s + 4.13·3-s − 2.69·4-s − 4.72·6-s − 7.54i·7-s + 7.64·8-s + 8.11·9-s − 18.6·11-s − 11.1·12-s − 16.9·13-s + 8.61i·14-s + 2.06·16-s − 2.51i·17-s − 9.26·18-s + (−8.62 + 16.9i)19-s + ⋯ |
| L(s) = 1 | − 0.570·2-s + 1.37·3-s − 0.674·4-s − 0.787·6-s − 1.07i·7-s + 0.955·8-s + 0.901·9-s − 1.69·11-s − 0.929·12-s − 1.30·13-s + 0.615i·14-s + 0.128·16-s − 0.148i·17-s − 0.514·18-s + (−0.454 + 0.890i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00762i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.00762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1869273402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1869273402\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + (8.62 - 16.9i)T \) |
| good | 2 | \( 1 + 1.14T + 4T^{2} \) |
| 3 | \( 1 - 4.13T + 9T^{2} \) |
| 7 | \( 1 + 7.54iT - 49T^{2} \) |
| 11 | \( 1 + 18.6T + 121T^{2} \) |
| 13 | \( 1 + 16.9T + 169T^{2} \) |
| 17 | \( 1 + 2.51iT - 289T^{2} \) |
| 23 | \( 1 - 41.9iT - 529T^{2} \) |
| 29 | \( 1 + 57.0iT - 841T^{2} \) |
| 31 | \( 1 - 30.5iT - 961T^{2} \) |
| 37 | \( 1 + 24.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 34.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 3.51iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 9.97iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 37.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 26.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 8.46T + 3.72e3T^{2} \) |
| 67 | \( 1 + 12.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 21.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 77.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 85.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 32.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 15.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 55.5T + 9.40e3T^{2} \) |
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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
−9.998970423454287136358362896354, −9.572140609964231403756879716140, −8.437920725274439793006365632591, −7.65369099051133233880799776760, −7.46188010583527781050935654239, −5.38863597287085857566250417950, −4.32017355446237678619494064941, −3.27480336031319048778954430717, −1.95858685704544978748932149795, −0.07282606060404208278609480675,
2.26495979214191281652415606985, 2.89361271270555547467285599488, 4.56531756202551800810791419946, 5.34684966324988953213205283039, 7.09766465869546148775809425426, 8.063616310497019440312200531683, 8.552237553733694306121443675766, 9.242289953925253305121991554853, 10.04592415743997640780755654515, 10.86715671649801118771338146793
