L(s) = 1 | + 3.16i·5-s + 7-s − 5.44i·11-s + 3.61i·13-s + 3.27·17-s − 3.20i·19-s − 0.673·23-s − 5.04·25-s − 2.85i·29-s − 3.71·31-s + 3.16i·35-s − 11.9i·37-s − 7.44·41-s − 12.5i·43-s + 4.06·47-s + ⋯ |
L(s) = 1 | + 1.41i·5-s + 0.377·7-s − 1.64i·11-s + 1.00i·13-s + 0.794·17-s − 0.735i·19-s − 0.140·23-s − 1.00·25-s − 0.529i·29-s − 0.667·31-s + 0.535i·35-s − 1.97i·37-s − 1.16·41-s − 1.91i·43-s + 0.592·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.525942373\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.525942373\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 3.16iT - 5T^{2} \) |
| 11 | \( 1 + 5.44iT - 11T^{2} \) |
| 13 | \( 1 - 3.61iT - 13T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 19 | \( 1 + 3.20iT - 19T^{2} \) |
| 23 | \( 1 + 0.673T + 23T^{2} \) |
| 29 | \( 1 + 2.85iT - 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 + 11.9iT - 37T^{2} \) |
| 41 | \( 1 + 7.44T + 41T^{2} \) |
| 43 | \( 1 + 12.5iT - 43T^{2} \) |
| 47 | \( 1 - 4.06T + 47T^{2} \) |
| 53 | \( 1 + 0.291iT - 53T^{2} \) |
| 59 | \( 1 - 0.0209iT - 59T^{2} \) |
| 61 | \( 1 + 5.34iT - 61T^{2} \) |
| 67 | \( 1 - 6.20iT - 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 1.35T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 13.6iT - 83T^{2} \) |
| 89 | \( 1 + 5.93T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−7.80284207360687434256114378521, −7.22169466933446873113013496619, −6.60673782674577104649775892287, −5.86768792973535299591023892578, −5.29871701385687679942346659620, −4.04352197871843801652988460482, −3.49809585644840302665824117977, −2.70359541068741930922731850300, −1.85131347334268533912870648477, −0.40227707903944276014061178720,
1.16199786682216951785504856091, 1.68329285250533953858609797135, 2.92552824878808738128140229109, 3.94110098616831913401292048524, 4.80011685095517154162436358429, 5.08637910644176595605978862474, 5.88005653621634258726932712870, 6.84647152245267091504138855503, 7.80959273678547931839015471176, 8.001770320026129584516770520742