| L(s) = 1 | − 3-s − 1.64i·5-s + i·7-s + 9-s + 0.797i·11-s + (−3.39 + 1.21i)13-s + 1.64i·15-s − 6.81·17-s + 3.89i·19-s − i·21-s + 3.97·23-s + 2.30·25-s − 27-s − 4.81·29-s − 0.797i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.733i·5-s + 0.377i·7-s + 0.333·9-s + 0.240i·11-s + (−0.941 + 0.338i)13-s + 0.423i·15-s − 1.65·17-s + 0.894i·19-s − 0.218i·21-s + 0.828·23-s + 0.461·25-s − 0.192·27-s − 0.894·29-s − 0.138i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.338 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9767870114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9767870114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (3.39 - 1.21i)T \) |
| good | 5 | \( 1 + 1.64iT - 5T^{2} \) |
| 11 | \( 1 - 0.797iT - 11T^{2} \) |
| 17 | \( 1 + 6.81T + 17T^{2} \) |
| 19 | \( 1 - 3.89iT - 19T^{2} \) |
| 23 | \( 1 - 3.97T + 23T^{2} \) |
| 29 | \( 1 + 4.81T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 0.466iT - 37T^{2} \) |
| 41 | \( 1 + 6.05iT - 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 4.01iT - 47T^{2} \) |
| 53 | \( 1 - 6.36T + 53T^{2} \) |
| 59 | \( 1 - 0.585iT - 59T^{2} \) |
| 61 | \( 1 - 3.45T + 61T^{2} \) |
| 67 | \( 1 + 0.150iT - 67T^{2} \) |
| 71 | \( 1 + 11.0iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 - 5.76iT - 83T^{2} \) |
| 89 | \( 1 - 4.51iT - 89T^{2} \) |
| 97 | \( 1 + 8.06iT - 97T^{2} \) |
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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
−8.299667606954482217486190207006, −7.34862744756907503890165545970, −6.83325672178050826590221429747, −5.91730654484640385566000901750, −5.21740708429652699578514353769, −4.60049112393333676655384701758, −3.89056700095742499327869074594, −2.52668107761440323780902855717, −1.73472929182097879647807829971, −0.38994291818306475658812832688,
0.817341441643520344941332362379, 2.30868601480218374299838328686, 2.95055453026034157661836145235, 4.12606280086333574959026329111, 4.76227765446774470761968480132, 5.57316544644773329493542975702, 6.47653918268973192413133772686, 7.04788775272588974545547555714, 7.45772730507682324709366041126, 8.561938997501273389909518186159
