| L(s) = 1 | − 0.198·2-s − 8.81·3-s − 7.96·4-s − 10.0·5-s + 1.75·6-s − 25.3·7-s + 3.17·8-s + 50.6·9-s + 2.00·10-s + 14.1·11-s + 70.1·12-s − 4.58·13-s + 5.03·14-s + 88.7·15-s + 63.0·16-s + 102.·17-s − 10.0·18-s − 69.2·19-s + 80.1·20-s + 223.·21-s − 2.81·22-s − 27.9·24-s − 23.6·25-s + 0.910·26-s − 208.·27-s + 201.·28-s + 210.·29-s + ⋯ |
| L(s) = 1 | − 0.0702·2-s − 1.69·3-s − 0.995·4-s − 0.900·5-s + 0.119·6-s − 1.36·7-s + 0.140·8-s + 1.87·9-s + 0.0632·10-s + 0.387·11-s + 1.68·12-s − 0.0977·13-s + 0.0961·14-s + 1.52·15-s + 0.985·16-s + 1.45·17-s − 0.131·18-s − 0.836·19-s + 0.895·20-s + 2.32·21-s − 0.0272·22-s − 0.237·24-s − 0.189·25-s + 0.00686·26-s − 1.48·27-s + 1.36·28-s + 1.35·29-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 | \( 1 + 0.198T + 8T^{2} \) |
| 3 | \( 1 + 8.81T + 27T^{2} \) |
| 5 | \( 1 + 10.0T + 125T^{2} \) |
| 7 | \( 1 + 25.3T + 343T^{2} \) |
| 11 | \( 1 - 14.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.58T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.2T + 6.85e3T^{2} \) |
| 29 | \( 1 - 210.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 282.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 174.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 128.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 130.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 96.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 276.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 458.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 350.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 765.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 906.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 448.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 430.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 334.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 851.T + 9.12e5T^{2} \) |
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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
−10.05246532803311819420322136102, −9.406765390249783911128152081080, −8.135004573024449769081146173114, −7.08237974716549123427103947894, −6.16566017111255261754469690073, −5.37591182666717376769233857387, −4.28685760616448520512851666308, −3.50757774283931798883446351487, −0.848375145528095318205229227218, 0,
0.848375145528095318205229227218, 3.50757774283931798883446351487, 4.28685760616448520512851666308, 5.37591182666717376769233857387, 6.16566017111255261754469690073, 7.08237974716549123427103947894, 8.135004573024449769081146173114, 9.406765390249783911128152081080, 10.05246532803311819420322136102
