| L(s) = 1 | + 5.41·2-s + 21.3·4-s + 7·7-s + 72.0·8-s + 52.2·11-s − 30.6·13-s + 37.8·14-s + 219.·16-s + 37.2·17-s + 80.2·19-s + 282.·22-s + 25.8·23-s − 165.·26-s + 149.·28-s − 20.9·29-s − 314.·31-s + 613.·32-s + 201.·34-s − 197.·37-s + 434.·38-s − 11.3·41-s + 33.8·43-s + 1.11e3·44-s + 139.·46-s − 361.·47-s + 49·49-s − 653.·52-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 2.66·4-s + 0.377·7-s + 3.18·8-s + 1.43·11-s − 0.654·13-s + 0.723·14-s + 3.43·16-s + 0.531·17-s + 0.968·19-s + 2.74·22-s + 0.234·23-s − 1.25·26-s + 1.00·28-s − 0.134·29-s − 1.82·31-s + 3.38·32-s + 1.01·34-s − 0.875·37-s + 1.85·38-s − 0.0432·41-s + 0.119·43-s + 3.81·44-s + 0.448·46-s − 1.12·47-s + 0.142·49-s − 1.74·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.758534075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.758534075\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 5.41T + 8T^{2} \) |
| 11 | \( 1 - 52.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 80.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 25.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 20.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 314.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 197.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 11.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 33.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 361.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 153.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 616T + 2.05e5T^{2} \) |
| 61 | \( 1 - 15.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 166.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 952T + 3.57e5T^{2} \) |
| 73 | \( 1 - 148.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 857.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 660.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 45.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.68e3T + 9.12e5T^{2} \) |
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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
−9.137782411391100420742336774239, −7.84870358613648712482921814105, −7.09503321598802074787734856127, −6.51244894676435169638107792322, −5.43981683249777740638572004133, −5.04026255138158721014435521234, −3.89834350273012229687821583878, −3.44528785216566537713760772089, −2.21941165757773480385632616640, −1.28656423209930191114592692801,
1.28656423209930191114592692801, 2.21941165757773480385632616640, 3.44528785216566537713760772089, 3.89834350273012229687821583878, 5.04026255138158721014435521234, 5.43981683249777740638572004133, 6.51244894676435169638107792322, 7.09503321598802074787734856127, 7.84870358613648712482921814105, 9.137782411391100420742336774239
