| L(s) = 1 | − 0.321i·3-s + (−2.10 − 0.742i)5-s − i·7-s + 2.89·9-s − 4.37·11-s + 5.86i·13-s + (−0.238 + 0.678i)15-s − 4.85i·17-s + 7.75·19-s − 0.321·21-s − 1.35i·23-s + (3.89 + 3.13i)25-s − 1.89i·27-s − 0.539·29-s − 2.97·31-s + ⋯ |
| L(s) = 1 | − 0.185i·3-s + (−0.943 − 0.332i)5-s − 0.377i·7-s + 0.965·9-s − 1.32·11-s + 1.62i·13-s + (−0.0616 + 0.175i)15-s − 1.17i·17-s + 1.77·19-s − 0.0701·21-s − 0.282i·23-s + (0.779 + 0.626i)25-s − 0.364i·27-s − 0.100·29-s − 0.533·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.332 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.049412691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.049412691\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.10 + 0.742i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + 0.321iT - 3T^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 - 5.86iT - 13T^{2} \) |
| 17 | \( 1 + 4.85iT - 17T^{2} \) |
| 19 | \( 1 - 7.75T + 19T^{2} \) |
| 23 | \( 1 + 1.35iT - 23T^{2} \) |
| 29 | \( 1 + 0.539T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 + 6.26iT - 37T^{2} \) |
| 41 | \( 1 - 2.64T + 41T^{2} \) |
| 43 | \( 1 + 4.64iT - 43T^{2} \) |
| 47 | \( 1 + 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 0.477iT - 53T^{2} \) |
| 59 | \( 1 + 7.75T + 59T^{2} \) |
| 61 | \( 1 + 7.57T + 61T^{2} \) |
| 67 | \( 1 + 3.79iT - 67T^{2} \) |
| 71 | \( 1 + 9.23T + 71T^{2} \) |
| 73 | \( 1 + 0.477iT - 73T^{2} \) |
| 79 | \( 1 + 1.88T + 79T^{2} \) |
| 83 | \( 1 + 15.2iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 13.6iT - 97T^{2} \) |
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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
−8.891609214735501094737749144042, −7.65301626789833673999836110496, −7.44275600592619632991540869903, −6.81484293683103619080119207943, −5.42296133013026099105168879117, −4.72656478661660463848272565132, −4.00909276473609591919401010680, −3.00400838523454663186236335394, −1.71691289004744151308542182241, −0.40752295417172116218769531437,
1.20198387760098223908542902484, 2.85037266762743988686316032547, 3.35739887604397945166297057520, 4.46716423360090988013916020381, 5.28125119537760535609299175065, 6.01909972045047645053414105182, 7.24616682041840676071444366595, 7.81576409539707384979414622420, 8.151294337351490407486250782521, 9.357381202149601716972571935712
