| L(s) = 1 | + (−1.02 + 1.71i)2-s + (1.02 − 2.81i)3-s + (−1.88 − 3.52i)4-s + (−3.20 + 3.83i)5-s + (3.77 + 4.66i)6-s + (6.55 − 6.55i)7-s + (7.98 + 0.400i)8-s + (−6.89 − 5.78i)9-s + (−3.27 − 9.44i)10-s + (−12.5 − 9.12i)11-s + (−11.8 + 1.68i)12-s + (−3.29 + 20.8i)13-s + (4.49 + 17.9i)14-s + (7.52 + 12.9i)15-s + (−8.90 + 13.2i)16-s + (−8.46 − 16.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.514 + 0.857i)2-s + (0.342 − 0.939i)3-s + (−0.470 − 0.882i)4-s + (−0.641 + 0.767i)5-s + (0.629 + 0.776i)6-s + (0.936 − 0.936i)7-s + (0.998 + 0.0501i)8-s + (−0.766 − 0.642i)9-s + (−0.327 − 0.944i)10-s + (−1.14 − 0.829i)11-s + (−0.990 + 0.140i)12-s + (−0.253 + 1.60i)13-s + (0.321 + 1.28i)14-s + (0.501 + 0.865i)15-s + (−0.556 + 0.830i)16-s + (−0.497 − 0.977i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.681 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.175796 - 0.403673i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.175796 - 0.403673i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.02 - 1.71i)T \) |
| 3 | \( 1 + (-1.02 + 2.81i)T \) |
| 5 | \( 1 + (3.20 - 3.83i)T \) |
| good | 7 | \( 1 + (-6.55 + 6.55i)T - 49iT^{2} \) |
| 11 | \( 1 + (12.5 + 9.12i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (3.29 - 20.8i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (8.46 + 16.6i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (2.97 - 9.15i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (5.96 - 0.944i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (1.86 + 5.73i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (11.8 + 3.84i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (58.0 + 9.19i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (35.5 + 48.9i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (29.8 + 29.8i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-18.0 + 35.4i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (0.720 - 1.41i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (0.672 + 0.925i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-84.4 - 61.3i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (10.6 + 20.8i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-36.5 - 112. i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (5.69 - 0.901i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (12.2 + 37.6i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-27.7 - 54.4i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-110. - 80.0i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-15.0 + 29.5i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−11.13038900905592319039629972144, −10.27941945079227532539087896653, −8.817416260127825528315293275921, −8.100557123949136492373231329571, −7.19555533751893888878131633422, −6.85128113277889364513046183387, −5.34774409204129097386683805706, −3.93464916404683395976426304730, −2.04122346815863678649304586295, −0.23222839402698633104127457808,
2.08294228309476974022996078107, 3.33963224369411663892057971017, 4.77427055153293719250149107222, 5.18611943198838733090655845664, 7.84479127344952483523025922436, 8.222476913781949039037796414042, 8.998990931294479983045972713459, 10.12963333523532872980098759134, 10.78537926429495839117773749590, 11.71142301766884460722981741595
