L(s) = 1 | + 3.50·2-s − 1.73i·3-s + 8.27·4-s + 2.23i·5-s − 6.06i·6-s + (−6.69 − 2.03i)7-s + 14.9·8-s − 2.99·9-s + 7.83i·10-s − 2.03·11-s − 14.3i·12-s + 18.0i·13-s + (−23.4 − 7.13i)14-s + 3.87·15-s + 19.3·16-s − 1.07i·17-s + ⋯ |
L(s) = 1 | + 1.75·2-s − 0.577i·3-s + 2.06·4-s + 0.447i·5-s − 1.01i·6-s + (−0.956 − 0.290i)7-s + 1.87·8-s − 0.333·9-s + 0.783i·10-s − 0.184·11-s − 1.19i·12-s + 1.38i·13-s + (−1.67 − 0.509i)14-s + 0.258·15-s + 1.21·16-s − 0.0631i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.02488 - 0.449624i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.02488 - 0.449624i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 + (6.69 + 2.03i)T \) |
good | 2 | \( 1 - 3.50T + 4T^{2} \) |
| 11 | \( 1 + 2.03T + 121T^{2} \) |
| 13 | \( 1 - 18.0iT - 169T^{2} \) |
| 17 | \( 1 + 1.07iT - 289T^{2} \) |
| 19 | \( 1 + 28.7iT - 361T^{2} \) |
| 23 | \( 1 - 24.8T + 529T^{2} \) |
| 29 | \( 1 + 38.4T + 841T^{2} \) |
| 31 | \( 1 - 44.0iT - 961T^{2} \) |
| 37 | \( 1 - 37.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 49.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 9.58T + 1.84e3T^{2} \) |
| 47 | \( 1 + 55.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 57.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 31.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 95.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 25.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 95.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 28.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 103. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 29.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 67.6iT - 9.40e3T^{2} \) |
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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
−13.42313435107440278352923778511, −12.79275707997601444624406076397, −11.68576554982934009329345627547, −10.88204949766889923834702905603, −9.192224632605330162802211565691, −6.96259081073287622452861910324, −6.78382406646877549013185762351, −5.27254538610868259803939914175, −3.78747182477589951776794821808, −2.50629075228292060907591811509,
2.89988737057636046266227934663, 3.99158677212916992690628271758, 5.40609977319768783533801597773, 6.08309355905801837521761843714, 7.78353005969709190664887274379, 9.500541686157581808648010977118, 10.67037686690589853454261874385, 11.85389232128567032967592811420, 12.91491561907220389357491343904, 13.22620749833861708032850748650