| 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
| \(L(2)\) |
\(\approx\) |
\(1.232453426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.232453426\) |
| \(L(\frac{5}{2})\) |
|
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
−9.202968378233252659141704051512, −8.533834945411762846597861024058, −7.59697862962240258030179655686, −6.66330468298484935098906007178, −6.03907608611185623656508084333, −4.90759975875999921035789430151, −3.91547437612357751692438509793, −3.12359798636333945384031977055, −1.56321108932515737373172477577, −0.64512935877119061700305049399,
0.64512935877119061700305049399, 1.56321108932515737373172477577, 3.12359798636333945384031977055, 3.91547437612357751692438509793, 4.90759975875999921035789430151, 6.03907608611185623656508084333, 6.66330468298484935098906007178, 7.59697862962240258030179655686, 8.533834945411762846597861024058, 9.202968378233252659141704051512
