| L(s) = 1 | − 0.110·2-s + 4.20·3-s − 7.98·4-s − 3.45·5-s − 0.464·6-s − 7.85·7-s + 1.76·8-s − 9.33·9-s + 0.381·10-s + 35.6·11-s − 33.5·12-s + 0.661·13-s + 0.867·14-s − 14.5·15-s + 63.7·16-s + 114.·17-s + 1.03·18-s + 71.2·19-s + 27.5·20-s − 33.0·21-s − 3.94·22-s − 23·23-s + 7.42·24-s − 113.·25-s − 0.0731·26-s − 152.·27-s + 62.7·28-s + ⋯ |
| L(s) = 1 | − 0.0390·2-s + 0.808·3-s − 0.998·4-s − 0.308·5-s − 0.0315·6-s − 0.424·7-s + 0.0780·8-s − 0.345·9-s + 0.0120·10-s + 0.978·11-s − 0.807·12-s + 0.0141·13-s + 0.0165·14-s − 0.249·15-s + 0.995·16-s + 1.63·17-s + 0.0135·18-s + 0.860·19-s + 0.308·20-s − 0.343·21-s − 0.0381·22-s − 0.208·23-s + 0.0631·24-s − 0.904·25-s − 0.000551·26-s − 1.08·27-s + 0.423·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 0.110T + 8T^{2} \) |
| 3 | \( 1 - 4.20T + 27T^{2} \) |
| 5 | \( 1 + 3.45T + 125T^{2} \) |
| 7 | \( 1 + 7.85T + 343T^{2} \) |
| 11 | \( 1 - 35.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.661T + 2.19e3T^{2} \) |
| 17 | \( 1 - 114.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.2T + 6.85e3T^{2} \) |
| 31 | \( 1 + 236.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 205.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 286.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 440.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 56.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 234.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 332.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 145.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 82.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 761.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 286.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 60.2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 723.T + 9.12e5T^{2} \) |
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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
−9.598371732520753885472652602989, −8.831167224922459888993453804074, −8.113746535208248200632435654262, −7.31512664206510589068260412391, −5.95270908178362987306935393592, −5.05614729340581602154541832024, −3.54668017728864139061730616215, −3.44873952161944721381974605915, −1.51251986487181539634589448456, 0,
1.51251986487181539634589448456, 3.44873952161944721381974605915, 3.54668017728864139061730616215, 5.05614729340581602154541832024, 5.95270908178362987306935393592, 7.31512664206510589068260412391, 8.113746535208248200632435654262, 8.831167224922459888993453804074, 9.598371732520753885472652602989
