| L(s) = 1 | + 1.73·2-s + 1.00·4-s − 0.674·5-s + 3.57·7-s − 1.72·8-s − 1.16·10-s + 2.26·11-s − 0.0370·13-s + 6.20·14-s − 4.99·16-s − 3.16·19-s − 0.676·20-s + 3.92·22-s − 3.82·23-s − 4.54·25-s − 0.0641·26-s + 3.58·28-s − 7.83·29-s − 9.64·31-s − 5.20·32-s − 2.41·35-s + 2.75·37-s − 5.47·38-s + 1.16·40-s − 4.21·41-s − 2.65·43-s + 2.26·44-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.501·4-s − 0.301·5-s + 1.35·7-s − 0.611·8-s − 0.369·10-s + 0.682·11-s − 0.0102·13-s + 1.65·14-s − 1.24·16-s − 0.725·19-s − 0.151·20-s + 0.836·22-s − 0.798·23-s − 0.908·25-s − 0.0125·26-s + 0.678·28-s − 1.45·29-s − 1.73·31-s − 0.920·32-s − 0.408·35-s + 0.452·37-s − 0.888·38-s + 0.184·40-s − 0.657·41-s − 0.404·43-s + 0.342·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 + 0.674T + 5T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 + 0.0370T + 13T^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 + 3.82T + 23T^{2} \) |
| 29 | \( 1 + 7.83T + 29T^{2} \) |
| 31 | \( 1 + 9.64T + 31T^{2} \) |
| 37 | \( 1 - 2.75T + 37T^{2} \) |
| 41 | \( 1 + 4.21T + 41T^{2} \) |
| 43 | \( 1 + 2.65T + 43T^{2} \) |
| 47 | \( 1 - 6.51T + 47T^{2} \) |
| 53 | \( 1 - 0.0248T + 53T^{2} \) |
| 59 | \( 1 + 4.54T + 59T^{2} \) |
| 61 | \( 1 + 0.235T + 61T^{2} \) |
| 67 | \( 1 + 2.75T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 3.39T + 79T^{2} \) |
| 83 | \( 1 + 8.75T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−7.51122415441980015282935926447, −6.64448734203619844410206566361, −5.83915190646267378736526870172, −5.37236477862228936779206784960, −4.57017417551962938458719075558, −3.99034967704808123450811389156, −3.54218893578909179334142909627, −2.25288645173641097167328586203, −1.64219736243972341659803653282, 0,
1.64219736243972341659803653282, 2.25288645173641097167328586203, 3.54218893578909179334142909627, 3.99034967704808123450811389156, 4.57017417551962938458719075558, 5.37236477862228936779206784960, 5.83915190646267378736526870172, 6.64448734203619844410206566361, 7.51122415441980015282935926447
