| L(s) = 1 | − 5.24·2-s + 4.66·3-s + 19.4·4-s + 9.98·5-s − 24.4·6-s − 16.6·7-s − 60.0·8-s − 5.22·9-s − 52.3·10-s − 35.9·11-s + 90.8·12-s + 36.7·13-s + 87.3·14-s + 46.5·15-s + 159.·16-s + 6.49·17-s + 27.3·18-s + 86.3·19-s + 194.·20-s − 77.7·21-s + 188.·22-s + 23·23-s − 280.·24-s − 25.3·25-s − 192.·26-s − 150.·27-s − 324.·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.897·3-s + 2.43·4-s + 0.892·5-s − 1.66·6-s − 0.900·7-s − 2.65·8-s − 0.193·9-s − 1.65·10-s − 0.984·11-s + 2.18·12-s + 0.784·13-s + 1.66·14-s + 0.801·15-s + 2.48·16-s + 0.0927·17-s + 0.358·18-s + 1.04·19-s + 2.17·20-s − 0.808·21-s + 1.82·22-s + 0.208·23-s − 2.38·24-s − 0.202·25-s − 1.45·26-s − 1.07·27-s − 2.18·28-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 | \( 1 + 5.24T + 8T^{2} \) |
| 3 | \( 1 - 4.66T + 27T^{2} \) |
| 5 | \( 1 - 9.98T + 125T^{2} \) |
| 7 | \( 1 + 16.6T + 343T^{2} \) |
| 11 | \( 1 + 35.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.49T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.3T + 6.85e3T^{2} \) |
| 31 | \( 1 - 0.154T + 2.97e4T^{2} \) |
| 37 | \( 1 - 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 146.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 39.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 11.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 285.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 27.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 19.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 741.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 329.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 556.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 240.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 117.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 453.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 427.T + 9.12e5T^{2} \) |
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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
−9.545108371750949255966154774042, −8.986923835633695584812862907078, −8.142634957189517257632754834132, −7.46371449522730547508748250128, −6.38246964907268822601027314191, −5.61020084572698806907564123320, −3.26870207896157347069887462374, −2.58978529099783537883091630689, −1.48068847445515176099299008208, 0,
1.48068847445515176099299008208, 2.58978529099783537883091630689, 3.26870207896157347069887462374, 5.61020084572698806907564123320, 6.38246964907268822601027314191, 7.46371449522730547508748250128, 8.142634957189517257632754834132, 8.986923835633695584812862907078, 9.545108371750949255966154774042
