| L(s) = 1 | + (−1.61 + 0.618i)3-s + (0.381 + 2.61i)7-s + (2.23 − 2.00i)9-s − 0.763i·11-s + 1.23i·13-s − 4.47·17-s − 7.23i·19-s + (−2.23 − 4i)21-s − 7.70i·23-s + (−2.38 + 4.61i)27-s + 4i·29-s + 3.23i·31-s + (0.472 + 1.23i)33-s − 4.47·37-s + (−0.763 − 2.00i)39-s + ⋯ |
| L(s) = 1 | + (−0.934 + 0.356i)3-s + (0.144 + 0.989i)7-s + (0.745 − 0.666i)9-s − 0.230i·11-s + 0.342i·13-s − 1.08·17-s − 1.66i·19-s + (−0.487 − 0.872i)21-s − 1.60i·23-s + (−0.458 + 0.888i)27-s + 0.742i·29-s + 0.581i·31-s + (0.0821 + 0.215i)33-s − 0.735·37-s + (−0.122 − 0.320i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8163396270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8163396270\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.61 - 0.618i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.381 - 2.61i)T \) |
| good | 11 | \( 1 + 0.763iT - 11T^{2} \) |
| 13 | \( 1 - 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 7.23iT - 19T^{2} \) |
| 23 | \( 1 + 7.70iT - 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 3.23iT - 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 - 3.23T + 47T^{2} \) |
| 53 | \( 1 + 9.23iT - 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 4.94iT - 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 12.1iT - 71T^{2} \) |
| 73 | \( 1 + 6.76iT - 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 7.23T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 14.1iT - 97T^{2} \) |
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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
−8.888252870865168735430367294574, −8.564904972544385185852518611929, −7.04691225749923885595152150451, −6.63955837358218650738017122754, −5.76125100715861922226080258357, −4.90378206557587266961397513436, −4.41155848360113745784466762464, −3.06128127042124562001491501700, −2.00370046655772203039439130270, −0.37522668597895561859101613654,
1.07276819059625396837706510844, 2.09012959192977059525731073856, 3.71651721174080239674426289323, 4.34775873897108425031676013372, 5.39205856383855464260407131328, 6.03915487913356727881254102953, 6.94927895544780034262113388303, 7.56875947588448571326806763758, 8.180060798442085534921625920486, 9.438737783910135746735864769967
