| L(s) = 1 | + 1.11·5-s + (1.37 − 2.26i)7-s + 0.812·11-s + 2·13-s + 3.55i·17-s + 4.52i·19-s + 3.63i·23-s − 3.75·25-s + 8.95i·29-s + 5.08·31-s + (1.53 − 2.52i)35-s + 0.718i·37-s + 10.4i·41-s − 1.11·43-s − 8.55·47-s + ⋯ |
| L(s) = 1 | + 0.498·5-s + (0.518 − 0.854i)7-s + 0.245·11-s + 0.554·13-s + 0.862i·17-s + 1.03i·19-s + 0.758i·23-s − 0.751·25-s + 1.66i·29-s + 0.914·31-s + (0.258 − 0.426i)35-s + 0.118i·37-s + 1.63i·41-s − 0.170·43-s − 1.24·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.192779609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192779609\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.37 + 2.26i)T \) |
| good | 5 | \( 1 - 1.11T + 5T^{2} \) |
| 11 | \( 1 - 0.812T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.55iT - 17T^{2} \) |
| 19 | \( 1 - 4.52iT - 19T^{2} \) |
| 23 | \( 1 - 3.63iT - 23T^{2} \) |
| 29 | \( 1 - 8.95iT - 29T^{2} \) |
| 31 | \( 1 - 5.08T + 31T^{2} \) |
| 37 | \( 1 - 0.718iT - 37T^{2} \) |
| 41 | \( 1 - 10.4iT - 41T^{2} \) |
| 43 | \( 1 + 1.11T + 43T^{2} \) |
| 47 | \( 1 + 8.55T + 47T^{2} \) |
| 53 | \( 1 - 5.08iT - 53T^{2} \) |
| 59 | \( 1 + 2.23iT - 59T^{2} \) |
| 61 | \( 1 - 5.27T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 5.87iT - 71T^{2} \) |
| 73 | \( 1 - 3.86iT - 73T^{2} \) |
| 79 | \( 1 + 1.70iT - 79T^{2} \) |
| 83 | \( 1 + 7.27iT - 83T^{2} \) |
| 89 | \( 1 + 3.37iT - 89T^{2} \) |
| 97 | \( 1 + 8.55iT - 97T^{2} \) |
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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
−8.284866758941006331657743940116, −8.012692284569818276896321134092, −7.00627248253799946341165843911, −6.33276978066671979588501848915, −5.61973879954459401447360442021, −4.75595488412385205166864138548, −3.89609003972498988277714909369, −3.24298680122825870865664901780, −1.78420114939453721766487060887, −1.24307902619873227175361245883,
0.66263600220072627153265327579, 2.05502858326835760358167445805, 2.57867399557314540475117732927, 3.75893271330754917427373112301, 4.73139542284908317273848814685, 5.34285607540264169590813804472, 6.18044338285903302872322598294, 6.71857669174164739440517900704, 7.75283410252268018751986743211, 8.407021964887668924104697709161
