| L(s) = 1 | + 0.447·2-s + 3·3-s − 7.79·4-s + 1.93·5-s + 1.34·6-s − 8.14·7-s − 7.06·8-s + 9·9-s + 0.863·10-s + 8.40·11-s − 23.3·12-s − 3.64·14-s + 5.79·15-s + 59.2·16-s − 52.1·17-s + 4.02·18-s + 48.8·19-s − 15.0·20-s − 24.4·21-s + 3.76·22-s + 88.9·23-s − 21.2·24-s − 121.·25-s + 27·27-s + 63.5·28-s + 191.·29-s + 2.59·30-s + ⋯ |
| L(s) = 1 | + 0.158·2-s + 0.577·3-s − 0.974·4-s + 0.172·5-s + 0.0913·6-s − 0.439·7-s − 0.312·8-s + 0.333·9-s + 0.0273·10-s + 0.230·11-s − 0.562·12-s − 0.0695·14-s + 0.0997·15-s + 0.925·16-s − 0.743·17-s + 0.0527·18-s + 0.589·19-s − 0.168·20-s − 0.253·21-s + 0.0364·22-s + 0.806·23-s − 0.180·24-s − 0.970·25-s + 0.192·27-s + 0.428·28-s + 1.22·29-s + 0.0157·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.966707389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966707389\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.447T + 8T^{2} \) |
| 5 | \( 1 - 1.93T + 125T^{2} \) |
| 7 | \( 1 + 8.14T + 343T^{2} \) |
| 11 | \( 1 - 8.40T + 1.33e3T^{2} \) |
| 17 | \( 1 + 52.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 88.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 136.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 436.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 202.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 618.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 453.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 500.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 886.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 123.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 673.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 681.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 939.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 754.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−10.21338397031554073697146290871, −9.494147467409624230886643425833, −8.849900984712230061402725236124, −7.978711443040905030979037819410, −6.85498327896399993110647949005, −5.75632467661409396278880192293, −4.63225841444495140501193452774, −3.73337949435757920908842646878, −2.59045115358214149536488077801, −0.846216138568199877910906498481,
0.846216138568199877910906498481, 2.59045115358214149536488077801, 3.73337949435757920908842646878, 4.63225841444495140501193452774, 5.75632467661409396278880192293, 6.85498327896399993110647949005, 7.978711443040905030979037819410, 8.849900984712230061402725236124, 9.494147467409624230886643425833, 10.21338397031554073697146290871
