| L(s) = 1 | − 0.685·2-s + 1.17·3-s − 1.53·4-s − 0.380·5-s − 0.804·6-s − 0.405·7-s + 2.41·8-s − 1.62·9-s + 0.261·10-s − 2.04·11-s − 1.79·12-s + 4.25·13-s + 0.277·14-s − 0.446·15-s + 1.40·16-s − 7.27·17-s + 1.11·18-s − 2.41·19-s + 0.582·20-s − 0.475·21-s + 1.40·22-s + 23-s + 2.83·24-s − 4.85·25-s − 2.91·26-s − 5.42·27-s + 0.620·28-s + ⋯ |
| L(s) = 1 | − 0.484·2-s + 0.677·3-s − 0.765·4-s − 0.170·5-s − 0.328·6-s − 0.153·7-s + 0.855·8-s − 0.541·9-s + 0.0825·10-s − 0.617·11-s − 0.518·12-s + 1.18·13-s + 0.0742·14-s − 0.115·15-s + 0.350·16-s − 1.76·17-s + 0.262·18-s − 0.553·19-s + 0.130·20-s − 0.103·21-s + 0.299·22-s + 0.208·23-s + 0.579·24-s − 0.970·25-s − 0.572·26-s − 1.04·27-s + 0.117·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.685T + 2T^{2} \) |
| 3 | \( 1 - 1.17T + 3T^{2} \) |
| 5 | \( 1 + 0.380T + 5T^{2} \) |
| 7 | \( 1 + 0.405T + 7T^{2} \) |
| 11 | \( 1 + 2.04T + 11T^{2} \) |
| 13 | \( 1 - 4.25T + 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 31 | \( 1 + 4.43T + 31T^{2} \) |
| 37 | \( 1 - 0.231T + 37T^{2} \) |
| 41 | \( 1 + 7.25T + 41T^{2} \) |
| 43 | \( 1 + 0.152T + 43T^{2} \) |
| 47 | \( 1 + 6.36T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 2.00T + 59T^{2} \) |
| 61 | \( 1 - 1.35T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 2.73T + 71T^{2} \) |
| 73 | \( 1 + 6.65T + 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 - 7.16T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−9.911077359118371293087360921633, −8.932676549745711657061153561983, −8.558531471221109662744894303031, −7.85388841130469070894272530882, −6.64054858472744822127881810082, −5.49817786333258185206072637370, −4.32946372985018360963263993345, −3.41869707072769676941446512973, −2.00534087450593736565947467881, 0,
2.00534087450593736565947467881, 3.41869707072769676941446512973, 4.32946372985018360963263993345, 5.49817786333258185206072637370, 6.64054858472744822127881810082, 7.85388841130469070894272530882, 8.558531471221109662744894303031, 8.932676549745711657061153561983, 9.911077359118371293087360921633
