| L(s) = 1 | + 1.86·2-s − 4.53·4-s − 3.46·5-s + 21.2·7-s − 23.3·8-s − 6.45·10-s − 54.1·11-s + 39.5·14-s − 7.16·16-s + 99.0·17-s + 64.1·19-s + 15.7·20-s − 100.·22-s + 153.·23-s − 112.·25-s − 96.2·28-s − 22.5·29-s − 241.·31-s + 173.·32-s + 184.·34-s − 73.5·35-s + 34.5·37-s + 119.·38-s + 80.8·40-s − 117.·41-s − 101.·43-s + 245.·44-s + ⋯ |
| L(s) = 1 | + 0.658·2-s − 0.566·4-s − 0.310·5-s + 1.14·7-s − 1.03·8-s − 0.204·10-s − 1.48·11-s + 0.754·14-s − 0.111·16-s + 1.41·17-s + 0.774·19-s + 0.175·20-s − 0.977·22-s + 1.39·23-s − 0.903·25-s − 0.649·28-s − 0.144·29-s − 1.39·31-s + 0.957·32-s + 0.930·34-s − 0.355·35-s + 0.153·37-s + 0.509·38-s + 0.319·40-s − 0.448·41-s − 0.359·43-s + 0.841·44-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 | \( 1 - 1.86T + 8T^{2} \) |
| 5 | \( 1 + 3.46T + 125T^{2} \) |
| 7 | \( 1 - 21.2T + 343T^{2} \) |
| 11 | \( 1 + 54.1T + 1.33e3T^{2} \) |
| 17 | \( 1 - 99.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 153.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 22.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 34.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 117.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 451.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 6.41T + 1.48e5T^{2} \) |
| 59 | \( 1 + 303.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 622.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 289.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 949.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 56.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 968.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 480.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 240.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.87e3T + 9.12e5T^{2} \) |
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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
−8.611521438209526360233998426329, −7.77565205157558580376535651418, −7.39341897648193401037953978227, −5.73708977022143559301029278048, −5.31438627317517731278103239725, −4.66096063401493582973729943321, −3.59652537551030676728213385818, −2.77921749273255429993820597747, −1.31812466557117650968130118127, 0,
1.31812466557117650968130118127, 2.77921749273255429993820597747, 3.59652537551030676728213385818, 4.66096063401493582973729943321, 5.31438627317517731278103239725, 5.73708977022143559301029278048, 7.39341897648193401037953978227, 7.77565205157558580376535651418, 8.611521438209526360233998426329
