| L(s) = 1 | − 2.44·2-s + 3-s + 3.99·4-s − 2.44·5-s − 2.44·6-s + 7-s − 4.89·8-s + 9-s + 5.99·10-s + 3.99·12-s + 6.89·13-s − 2.44·14-s − 2.44·15-s + 3.99·16-s − 0.550·17-s − 2.44·18-s − 0.449·19-s − 9.79·20-s + 21-s + 6·23-s − 4.89·24-s + 0.999·25-s − 16.8·26-s + 27-s + 3.99·28-s + 1.89·29-s + 5.99·30-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 0.577·3-s + 1.99·4-s − 1.09·5-s − 0.999·6-s + 0.377·7-s − 1.73·8-s + 0.333·9-s + 1.89·10-s + 1.15·12-s + 1.91·13-s − 0.654·14-s − 0.632·15-s + 0.999·16-s − 0.133·17-s − 0.577·18-s − 0.103·19-s − 2.19·20-s + 0.218·21-s + 1.25·23-s − 0.999·24-s + 0.199·25-s − 3.31·26-s + 0.192·27-s + 0.755·28-s + 0.352·29-s + 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9103031107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9103031107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 17 | \( 1 + 0.550T + 17T^{2} \) |
| 19 | \( 1 + 0.449T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 - 0.550T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 7.89T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 + 0.449T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 8.44T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 5.89T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.582961843504661814972934814529, −8.333854013153734751896655491834, −7.74851595444853039329574339737, −6.88236497167738781734271983423, −6.29492396907212322616902875580, −4.82794472560227298101268428262, −3.76272811892513590923826996963, −2.97568520213590850281150463075, −1.68441610522505223825387644787, −0.796797708794726098704515066891,
0.796797708794726098704515066891, 1.68441610522505223825387644787, 2.97568520213590850281150463075, 3.76272811892513590923826996963, 4.82794472560227298101268428262, 6.29492396907212322616902875580, 6.88236497167738781734271983423, 7.74851595444853039329574339737, 8.333854013153734751896655491834, 8.582961843504661814972934814529
