| L(s) = 1 | + 4i·3-s + 2·5-s − 8i·7-s − 7·9-s − 4i·11-s − 14·13-s + 8i·15-s + 18·17-s − 12i·19-s + 32·21-s + 40i·23-s − 21·25-s + 8i·27-s − 14·29-s − 32i·31-s + ⋯ |
| L(s) = 1 | + 1.33i·3-s + 0.400·5-s − 1.14i·7-s − 0.777·9-s − 0.363i·11-s − 1.07·13-s + 0.533i·15-s + 1.05·17-s − 0.631i·19-s + 1.52·21-s + 1.73i·23-s − 0.839·25-s + 0.296i·27-s − 0.482·29-s − 1.03i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.949204 + 0.393173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.949204 + 0.393173i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 4iT - 9T^{2} \) |
| 5 | \( 1 - 2T + 25T^{2} \) |
| 7 | \( 1 + 8iT - 49T^{2} \) |
| 11 | \( 1 + 4iT - 121T^{2} \) |
| 13 | \( 1 + 14T + 169T^{2} \) |
| 17 | \( 1 - 18T + 289T^{2} \) |
| 19 | \( 1 + 12iT - 361T^{2} \) |
| 23 | \( 1 - 40iT - 529T^{2} \) |
| 29 | \( 1 + 14T + 841T^{2} \) |
| 31 | \( 1 + 32iT - 961T^{2} \) |
| 37 | \( 1 + 30T + 1.36e3T^{2} \) |
| 41 | \( 1 + 14T + 1.68e3T^{2} \) |
| 43 | \( 1 - 28iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 16iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 66T + 2.80e3T^{2} \) |
| 59 | \( 1 + 52iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 82T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 56iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 66T + 5.32e3T^{2} \) |
| 79 | \( 1 + 16iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 140iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 30T + 7.92e3T^{2} \) |
| 97 | \( 1 + 14T + 9.40e3T^{2} \) |
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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
−16.77294987199429709208595907091, −15.62040707469741893079976141245, −14.47842636901374269930730860291, −13.36736872753758180151349422357, −11.47151455699479010753422613009, −10.17831541149061037532087029510, −9.520362719450938750303497734060, −7.48953961653248355907872725716, −5.31197217598345952792548097481, −3.75856410002964676917284222931,
2.21172993079294234645440893430, 5.57276917665285188694963222328, 7.01555529360505269127969166535, 8.399845673105813023778877332252, 10.00640413076516651543658495963, 12.16437619036055732214846434883, 12.43486426381544313179562067435, 13.95949811004052582952191382826, 15.02586619141541630290157328234, 16.69732549621509746387669958961
