L(s) = 1 | + 1.87·2-s − 1.77·3-s + 1.51·4-s + 5-s − 3.33·6-s − 4.25·7-s − 0.901·8-s + 0.159·9-s + 1.87·10-s − 2.70·12-s − 1.36·13-s − 7.98·14-s − 1.77·15-s − 4.73·16-s − 2.09·17-s + 0.299·18-s − 0.604·19-s + 1.51·20-s + 7.56·21-s − 4.39·23-s + 1.60·24-s + 25-s − 2.55·26-s + 5.04·27-s − 6.46·28-s + 6.63·29-s − 3.33·30-s + ⋯ |
L(s) = 1 | + 1.32·2-s − 1.02·3-s + 0.759·4-s + 0.447·5-s − 1.36·6-s − 1.60·7-s − 0.318·8-s + 0.0531·9-s + 0.593·10-s − 0.779·12-s − 0.377·13-s − 2.13·14-s − 0.458·15-s − 1.18·16-s − 0.508·17-s + 0.0705·18-s − 0.138·19-s + 0.339·20-s + 1.65·21-s − 0.916·23-s + 0.327·24-s + 0.200·25-s − 0.500·26-s + 0.971·27-s − 1.22·28-s + 1.23·29-s − 0.608·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 3 | \( 1 + 1.77T + 3T^{2} \) |
| 7 | \( 1 + 4.25T + 7T^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 + 2.09T + 17T^{2} \) |
| 19 | \( 1 + 0.604T + 19T^{2} \) |
| 23 | \( 1 + 4.39T + 23T^{2} \) |
| 29 | \( 1 - 6.63T + 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 + 7.40T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 + 6.65T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 67 | \( 1 - 9.86T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 - 0.722T + 73T^{2} \) |
| 79 | \( 1 + 5.67T + 79T^{2} \) |
| 83 | \( 1 - 0.952T + 83T^{2} \) |
| 89 | \( 1 - 1.24T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−10.27264911966906059073164078545, −9.667725079096374696400179137200, −8.532605363527969381480069835290, −6.67418225543118668489603830253, −6.47852259253136746386084737987, −5.59070383351561562682853477032, −4.78372736270906072334256046136, −3.60177501393623652959348241926, −2.58738402693973646110576449948, 0,
2.58738402693973646110576449948, 3.60177501393623652959348241926, 4.78372736270906072334256046136, 5.59070383351561562682853477032, 6.47852259253136746386084737987, 6.67418225543118668489603830253, 8.532605363527969381480069835290, 9.667725079096374696400179137200, 10.27264911966906059073164078545