| L(s) = 1 | − 3.45·2-s + 3.97·4-s − 7.89·5-s + 4.57·7-s + 13.9·8-s + 27.2·10-s + 7.16·11-s + 9.66·13-s − 15.8·14-s − 79.9·16-s + 69.1·17-s − 21.2·19-s − 31.3·20-s − 24.8·22-s − 23·23-s − 62.7·25-s − 33.4·26-s + 18.1·28-s − 39.5·29-s + 28.5·31-s + 165.·32-s − 239.·34-s − 36.1·35-s − 170.·37-s + 73.4·38-s − 110.·40-s − 395.·41-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.496·4-s − 0.705·5-s + 0.247·7-s + 0.616·8-s + 0.863·10-s + 0.196·11-s + 0.206·13-s − 0.302·14-s − 1.24·16-s + 0.987·17-s − 0.256·19-s − 0.350·20-s − 0.240·22-s − 0.208·23-s − 0.501·25-s − 0.252·26-s + 0.122·28-s − 0.253·29-s + 0.165·31-s + 0.912·32-s − 1.20·34-s − 0.174·35-s − 0.759·37-s + 0.313·38-s − 0.434·40-s − 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 3.45T + 8T^{2} \) |
| 5 | \( 1 + 7.89T + 125T^{2} \) |
| 7 | \( 1 - 4.57T + 343T^{2} \) |
| 11 | \( 1 - 7.16T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.66T + 2.19e3T^{2} \) |
| 17 | \( 1 - 69.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 21.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 39.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 28.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 170.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 395.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 214.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 387.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 268.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 552.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 354.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 293.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 505.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 38.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 111.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.47e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.27e3T + 9.12e5T^{2} \) |
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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
−11.29003242918109938118206321167, −10.30203812570783210904954493818, −9.462772578635787994738831930934, −8.323059042364131902074658833523, −7.82187972416958722246274901703, −6.64740333756136538911985580378, −4.98649323871346260357723225617, −3.61249529778295146506307446280, −1.57457788736723542944649945275, 0,
1.57457788736723542944649945275, 3.61249529778295146506307446280, 4.98649323871346260357723225617, 6.64740333756136538911985580378, 7.82187972416958722246274901703, 8.323059042364131902074658833523, 9.462772578635787994738831930934, 10.30203812570783210904954493818, 11.29003242918109938118206321167
