| L(s) = 1 | + 0.643i·2-s + 15.5·4-s − 12.3i·5-s + 75.5·7-s + 20.3i·8-s + 7.93·10-s + 34.0i·11-s − 129.·13-s + 48.5i·14-s + 236.·16-s + 332. i·17-s + 320.·19-s − 192. i·20-s − 21.9·22-s + 629. i·23-s + ⋯ |
| L(s) = 1 | + 0.160i·2-s + 0.974·4-s − 0.493i·5-s + 1.54·7-s + 0.317i·8-s + 0.0793·10-s + 0.281i·11-s − 0.768·13-s + 0.247i·14-s + 0.923·16-s + 1.15i·17-s + 0.886·19-s − 0.480i·20-s − 0.0452·22-s + 1.18i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.351496663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.351496663\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 453. iT \) |
| good | 2 | \( 1 - 0.643iT - 16T^{2} \) |
| 5 | \( 1 + 12.3iT - 625T^{2} \) |
| 7 | \( 1 - 75.5T + 2.40e3T^{2} \) |
| 11 | \( 1 - 34.0iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 129.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 332. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 320.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 629. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.27e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 163.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.02e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.33e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.67e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.98e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.94e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 + 6.08e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 3.11e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 4.16e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 7.58e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.05e4T + 3.89e7T^{2} \) |
| 83 | \( 1 - 4.10e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.06e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 794.T + 8.85e7T^{2} \) |
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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
−10.65491396633771862278847073396, −9.369160814957396689391264559918, −8.339997480147870112373455090407, −7.62628976487791373705427035854, −6.88973954741454011079590434622, −5.45577935586364304477277534248, −4.98622988239060684910416729860, −3.51732857335654570031882151817, −2.00703113798538616814750690220, −1.31158373648439051194654606250,
0.899590276727662110653486646428, 2.18294510315724998074691935807, 2.98045650049247762721179320976, 4.53461700198766840629064873968, 5.45005757230328992354656060561, 6.66277844294003644463083680468, 7.47044255279646546704089369759, 8.111696309847647870278861918235, 9.370731084260515037774025766693, 10.45756253855172318039911007187
