L(s) = 1 | + 2.64i·7-s + 3·9-s + 5.29·11-s + 5.29i·23-s − 5·25-s − 2i·29-s − 6i·37-s + 5.29·43-s − 7.00·49-s + 10i·53-s + 7.93i·63-s + 15.8·67-s + 5.29i·71-s + 14.0i·77-s + 15.8i·79-s + ⋯ |
L(s) = 1 | + 0.999i·7-s + 9-s + 1.59·11-s + 1.10i·23-s − 25-s − 0.371i·29-s − 0.986i·37-s + 0.806·43-s − 49-s + 1.37i·53-s + 0.999i·63-s + 1.93·67-s + 0.627i·71-s + 1.59i·77-s + 1.78i·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.026695214\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.026695214\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 5.29iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 - 5.29iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 15.8iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.474803968307863373775983600235, −8.740964683145984380508465141837, −7.76323678335916145494317881236, −6.98626934369784705581226205114, −6.15730352341338052470420320490, −5.44541766318167866861287528740, −4.26681220512043031696503215102, −3.65191511571204231747643109576, −2.26272384328818931489374653204, −1.29806902455609965355266323446,
0.891045092888046737593636769054, 1.89755061656783189113881381262, 3.51538272761900889906598995391, 4.13324079529668552505420176572, 4.85947198389806214212874203375, 6.27661407144869441251839686396, 6.75670853980951518865137403653, 7.50113652565358697706118520387, 8.370067422434350578144795520702, 9.321815729346054607004549193538