| L(s) = 1 | − 1.28·2-s + 3-s − 0.353·4-s + 0.613·5-s − 1.28·6-s − 0.715·7-s + 3.01·8-s + 9-s − 0.786·10-s − 0.353·12-s + 13-s + 0.918·14-s + 0.613·15-s − 3.16·16-s − 4.29·17-s − 1.28·18-s + 2.10·19-s − 0.216·20-s − 0.715·21-s + 7.24·23-s + 3.01·24-s − 4.62·25-s − 1.28·26-s + 27-s + 0.252·28-s − 1.84·29-s − 0.786·30-s + ⋯ |
| L(s) = 1 | − 0.907·2-s + 0.577·3-s − 0.176·4-s + 0.274·5-s − 0.523·6-s − 0.270·7-s + 1.06·8-s + 0.333·9-s − 0.248·10-s − 0.101·12-s + 0.277·13-s + 0.245·14-s + 0.158·15-s − 0.792·16-s − 1.04·17-s − 0.302·18-s + 0.483·19-s − 0.0484·20-s − 0.156·21-s + 1.51·23-s + 0.616·24-s − 0.924·25-s − 0.251·26-s + 0.192·27-s + 0.0477·28-s − 0.342·29-s − 0.143·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.293967687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.293967687\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.28T + 2T^{2} \) |
| 5 | \( 1 - 0.613T + 5T^{2} \) |
| 7 | \( 1 + 0.715T + 7T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 - 2.10T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 29 | \( 1 + 1.84T + 29T^{2} \) |
| 31 | \( 1 - 2.33T + 31T^{2} \) |
| 37 | \( 1 - 2.70T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 - 8.60T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 9.28T + 59T^{2} \) |
| 61 | \( 1 + 4.01T + 61T^{2} \) |
| 67 | \( 1 + 1.32T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 8.78T + 79T^{2} \) |
| 83 | \( 1 - 9.92T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.514719592396366519905387201539, −7.66092371623263005454605583611, −7.15269994364973322447399100055, −6.31480743993535900117707409242, −5.32560952155512495826583623084, −4.50094699879567322955609790503, −3.73644269099810428835058266620, −2.70681466836321375359069244329, −1.77484985011295274703729221519, −0.72232422602996075295780868103,
0.72232422602996075295780868103, 1.77484985011295274703729221519, 2.70681466836321375359069244329, 3.73644269099810428835058266620, 4.50094699879567322955609790503, 5.32560952155512495826583623084, 6.31480743993535900117707409242, 7.15269994364973322447399100055, 7.66092371623263005454605583611, 8.514719592396366519905387201539
