| L(s) = 1 | − 3.19·2-s + 1.55·3-s + 2.17·4-s − 19.2·5-s − 4.94·6-s + 19.8·7-s + 18.5·8-s − 24.5·9-s + 61.5·10-s − 17.2·11-s + 3.37·12-s + 37.7·13-s − 63.3·14-s − 29.9·15-s − 76.6·16-s + 22.1·17-s + 78.4·18-s − 6.74·19-s − 42.0·20-s + 30.7·21-s + 54.9·22-s + 28.7·24-s + 247.·25-s − 120.·26-s − 80.0·27-s + 43.2·28-s + 226.·29-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 0.298·3-s + 0.272·4-s − 1.72·5-s − 0.336·6-s + 1.07·7-s + 0.820·8-s − 0.910·9-s + 1.94·10-s − 0.472·11-s + 0.0812·12-s + 0.805·13-s − 1.20·14-s − 0.515·15-s − 1.19·16-s + 0.316·17-s + 1.02·18-s − 0.0814·19-s − 0.470·20-s + 0.319·21-s + 0.532·22-s + 0.244·24-s + 1.97·25-s − 0.908·26-s − 0.570·27-s + 0.292·28-s + 1.45·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 \) |
| good | 2 | \( 1 + 3.19T + 8T^{2} \) |
| 3 | \( 1 - 1.55T + 27T^{2} \) |
| 5 | \( 1 + 19.2T + 125T^{2} \) |
| 7 | \( 1 - 19.8T + 343T^{2} \) |
| 11 | \( 1 + 17.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 22.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.74T + 6.85e3T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 292.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 113.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 443.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 277.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 75.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 195.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 354.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 36.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 40.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 635.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 354.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 726.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 621.T + 9.12e5T^{2} \) |
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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
−9.988665723480719620978605737987, −8.527441625276860189947603964586, −8.346098013104595907489960771653, −7.899849872922649030608261062627, −6.76415751549436106615313296121, −5.05903462844826785749407528420, −4.19867765287552975628882551381, −2.94776648803762543601096327356, −1.19380709255711226744550255251, 0,
1.19380709255711226744550255251, 2.94776648803762543601096327356, 4.19867765287552975628882551381, 5.05903462844826785749407528420, 6.76415751549436106615313296121, 7.899849872922649030608261062627, 8.346098013104595907489960771653, 8.527441625276860189947603964586, 9.988665723480719620978605737987
