L(s) = 1 | − 1.99·2-s + 0.764·3-s + 1.98·4-s + 2.38·5-s − 1.52·6-s − 0.101·7-s + 0.0328·8-s − 2.41·9-s − 4.76·10-s − 3.79·11-s + 1.51·12-s − 3.75·13-s + 0.201·14-s + 1.82·15-s − 4.03·16-s − 4.24·17-s + 4.81·18-s − 3.15·19-s + 4.73·20-s − 0.0773·21-s + 7.56·22-s − 23-s + 0.0251·24-s + 0.689·25-s + 7.49·26-s − 4.14·27-s − 0.200·28-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.441·3-s + 0.991·4-s + 1.06·5-s − 0.623·6-s − 0.0382·7-s + 0.0116·8-s − 0.804·9-s − 1.50·10-s − 1.14·11-s + 0.438·12-s − 1.04·13-s + 0.0539·14-s + 0.471·15-s − 1.00·16-s − 1.03·17-s + 1.13·18-s − 0.723·19-s + 1.05·20-s − 0.0168·21-s + 1.61·22-s − 0.208·23-s + 0.00513·24-s + 0.137·25-s + 1.46·26-s − 0.797·27-s − 0.0378·28-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 | \( 1 + 1.99T + 2T^{2} \) |
| 3 | \( 1 - 0.764T + 3T^{2} \) |
| 5 | \( 1 - 2.38T + 5T^{2} \) |
| 7 | \( 1 + 0.101T + 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 31 | \( 1 - 0.905T + 31T^{2} \) |
| 37 | \( 1 - 6.40T + 37T^{2} \) |
| 41 | \( 1 + 3.03T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 6.38T + 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 + 5.17T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 6.65T + 73T^{2} \) |
| 79 | \( 1 - 7.81T + 79T^{2} \) |
| 83 | \( 1 + 9.08T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
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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
−9.715464427852257088161101214424, −9.417550508208852806244147186787, −8.391048336560780646785789593826, −7.85718928759285209390848775848, −6.76921118969476574965330549130, −5.74407961815169729440706545706, −4.62507173355967503109310822525, −2.64806495271161451677521196668, −2.05427596932475585611029400552, 0,
2.05427596932475585611029400552, 2.64806495271161451677521196668, 4.62507173355967503109310822525, 5.74407961815169729440706545706, 6.76921118969476574965330549130, 7.85718928759285209390848775848, 8.391048336560780646785789593826, 9.417550508208852806244147186787, 9.715464427852257088161101214424