| L(s) = 1 | − 3-s + (0.456 + 2.18i)5-s + 4.37i·7-s + 9-s − 5.58i·11-s + 4.37·13-s + (−0.456 − 2.18i)15-s + 5.58i·17-s + 4i·19-s − 4.37i·21-s + (−4.58 + 1.99i)25-s − 27-s − 2.55i·29-s − 5.29·31-s + 5.58i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (0.204 + 0.978i)5-s + 1.65i·7-s + 0.333·9-s − 1.68i·11-s + 1.21·13-s + (−0.117 − 0.565i)15-s + 1.35i·17-s + 0.917i·19-s − 0.955i·21-s + (−0.916 + 0.399i)25-s − 0.192·27-s − 0.473i·29-s − 0.950·31-s + 0.971i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.621326 + 1.00973i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.621326 + 1.00973i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-0.456 - 2.18i)T \) |
| good | 7 | \( 1 - 4.37iT - 7T^{2} \) |
| 11 | \( 1 + 5.58iT - 11T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 - 5.58iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2.55iT - 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 - 2.55T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 6.92iT - 47T^{2} \) |
| 53 | \( 1 + 7.84T + 53T^{2} \) |
| 59 | \( 1 + 1.58iT - 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 + 3.16T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 - 7.16T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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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
−10.53201732351973894668087574662, −9.466501796708246040028459884467, −8.539779000130337136313725372781, −8.007584301556959660728397407700, −6.44846866883470353043225374928, −5.94240905471771660069797967738, −5.62187640743578961084655255397, −3.83609255640133584770568582663, −3.01114756980248861315450659624, −1.69158491276635276746830240489,
0.61806894987921123948203027540, 1.73981160097029671869661774769, 3.67136262637149631619457737304, 4.64253745015148818033239094293, 5.06335143269824095529225077435, 6.49605042109952434212769258781, 7.16439590435401688890404723891, 7.899437481272890613249075894612, 9.184141004625997201111053704290, 9.719398087281507617051183315980
