| L(s) = 1 | − 7.67i·5-s + 7.21·7-s − 6.05i·11-s − 2.29·13-s − 21.8i·17-s + 34.8·19-s − 21.5i·23-s − 33.8·25-s + 10.9i·29-s − 37.6·31-s − 55.3i·35-s + 34.8·37-s − 13.3i·41-s + 60.5·43-s − 3.34i·47-s + ⋯ |
| L(s) = 1 | − 1.53i·5-s + 1.03·7-s − 0.550i·11-s − 0.176·13-s − 1.28i·17-s + 1.83·19-s − 0.935i·23-s − 1.35·25-s + 0.376i·29-s − 1.21·31-s − 1.58i·35-s + 0.941·37-s − 0.325i·41-s + 1.40·43-s − 0.0712i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.305225838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.305225838\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 7.67iT - 25T^{2} \) |
| 7 | \( 1 - 7.21T + 49T^{2} \) |
| 11 | \( 1 + 6.05iT - 121T^{2} \) |
| 13 | \( 1 + 2.29T + 169T^{2} \) |
| 17 | \( 1 + 21.8iT - 289T^{2} \) |
| 19 | \( 1 - 34.8T + 361T^{2} \) |
| 23 | \( 1 + 21.5iT - 529T^{2} \) |
| 29 | \( 1 - 10.9iT - 841T^{2} \) |
| 31 | \( 1 + 37.6T + 961T^{2} \) |
| 37 | \( 1 - 34.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 60.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 3.34iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 35.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 37.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 25.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 25.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 37.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 77.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 31.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 55.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 5.43iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 52.8T + 9.40e3T^{2} \) |
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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
−8.645026235153728798947539749288, −7.80297446445141003325342947477, −7.32647147089855694877226510398, −5.96350717234578812423270051450, −5.05742425887989121288885010174, −4.91858586134010518793292276367, −3.78837130754393743514771239389, −2.55049100114061369055955430629, −1.29487573534759250649344706111, −0.60996996095935543636524780147,
1.39829873711421742231565580609, 2.34043804638585382130870535047, 3.33172181967903098283051249924, 4.12740122827745703484983759229, 5.26322142722233833743805831527, 5.95792228165653810300202954131, 6.93063985349379842416891006810, 7.60402064885394821375728483443, 7.971019392860924201847649982998, 9.256427269810555792956701763438
