| L(s) = 1 | + 3.82·2-s + 6.63·4-s − 0.275·5-s + 0.0981·7-s − 5.20·8-s − 1.05·10-s + 0.749·11-s + 0.375·14-s − 73.0·16-s − 53.7·17-s + 145.·19-s − 1.82·20-s + 2.86·22-s − 29.3·23-s − 124.·25-s + 0.651·28-s + 267.·29-s − 51.5·31-s − 237.·32-s − 205.·34-s − 0.0270·35-s − 133.·37-s + 556.·38-s + 1.43·40-s − 430.·41-s − 282.·43-s + 4.97·44-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.829·4-s − 0.0246·5-s + 0.00529·7-s − 0.230·8-s − 0.0333·10-s + 0.0205·11-s + 0.00716·14-s − 1.14·16-s − 0.767·17-s + 1.75·19-s − 0.0204·20-s + 0.0278·22-s − 0.266·23-s − 0.999·25-s + 0.00439·28-s + 1.71·29-s − 0.298·31-s − 1.31·32-s − 1.03·34-s − 0.000130·35-s − 0.595·37-s + 2.37·38-s + 0.00566·40-s − 1.63·41-s − 1.00·43-s + 0.0170·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3.82T + 8T^{2} \) |
| 5 | \( 1 + 0.275T + 125T^{2} \) |
| 7 | \( 1 - 0.0981T + 343T^{2} \) |
| 11 | \( 1 - 0.749T + 1.33e3T^{2} \) |
| 17 | \( 1 + 53.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 145.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 29.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 267.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 51.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 430.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 282.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 212.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 573.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 495.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 310.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 103.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 203.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 685.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 636.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 506.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 700.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 874.T + 9.12e5T^{2} \) |
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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
−8.692176204565928599328534694260, −7.74213310776660035194132871131, −6.77393232678873264967984257800, −6.11488226354460000877418199998, −5.16306748807120864296095922091, −4.62313660602144270669175261102, −3.55738225100763759149951783788, −2.91092149370554596828966998316, −1.64014366068712475396512350575, 0,
1.64014366068712475396512350575, 2.91092149370554596828966998316, 3.55738225100763759149951783788, 4.62313660602144270669175261102, 5.16306748807120864296095922091, 6.11488226354460000877418199998, 6.77393232678873264967984257800, 7.74213310776660035194132871131, 8.692176204565928599328534694260
