| L(s) = 1 | + 0.282·2-s + 8.03·3-s − 7.91·4-s − 4.50·5-s + 2.27·6-s − 16.5·7-s − 4.50·8-s + 37.5·9-s − 1.27·10-s + 57.1·11-s − 63.6·12-s − 4.84·13-s − 4.69·14-s − 36.1·15-s + 62.0·16-s − 116.·17-s + 10.6·18-s + 27.0·19-s + 35.6·20-s − 133.·21-s + 16.1·22-s + 23·23-s − 36.1·24-s − 104.·25-s − 1.37·26-s + 84.7·27-s + 131.·28-s + ⋯ |
| L(s) = 1 | + 0.100·2-s + 1.54·3-s − 0.989·4-s − 0.402·5-s + 0.154·6-s − 0.895·7-s − 0.199·8-s + 1.39·9-s − 0.0402·10-s + 1.56·11-s − 1.53·12-s − 0.103·13-s − 0.0896·14-s − 0.622·15-s + 0.970·16-s − 1.66·17-s + 0.139·18-s + 0.326·19-s + 0.398·20-s − 1.38·21-s + 0.156·22-s + 0.208·23-s − 0.307·24-s − 0.837·25-s − 0.0103·26-s + 0.603·27-s + 0.887·28-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 | \( 1 - 0.282T + 8T^{2} \) |
| 3 | \( 1 - 8.03T + 27T^{2} \) |
| 5 | \( 1 + 4.50T + 125T^{2} \) |
| 7 | \( 1 + 16.5T + 343T^{2} \) |
| 11 | \( 1 - 57.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.84T + 2.19e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.0T + 6.85e3T^{2} \) |
| 31 | \( 1 + 93.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 238.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 263.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 65.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 382.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 19.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 220.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 438.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.05e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 340.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 531.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 112.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.88e3T + 9.12e5T^{2} \) |
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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
−9.168644849564871777546018078162, −9.127744728762102346027960223503, −8.209659933585092184235850638673, −7.17414124602659786569794318511, −6.26480319973153943741290703157, −4.65312316075907065889546784508, −3.78056894339320213618222802789, −3.26685741455872208892196200106, −1.76125702414499002806386271034, 0,
1.76125702414499002806386271034, 3.26685741455872208892196200106, 3.78056894339320213618222802789, 4.65312316075907065889546784508, 6.26480319973153943741290703157, 7.17414124602659786569794318511, 8.209659933585092184235850638673, 9.127744728762102346027960223503, 9.168644849564871777546018078162
