L(s) = 1 | + 1.33·2-s − 3-s − 0.226·4-s − 3.31·5-s − 1.33·6-s − 1.53·7-s − 2.96·8-s + 9-s − 4.41·10-s − 11-s + 0.226·12-s − 2.04·14-s + 3.31·15-s − 3.49·16-s + 5.96·17-s + 1.33·18-s + 6.46·19-s + 0.749·20-s + 1.53·21-s − 1.33·22-s + 5.32·23-s + 2.96·24-s + 5.97·25-s − 27-s + 0.347·28-s − 0.191·29-s + 4.41·30-s + ⋯ |
L(s) = 1 | + 0.941·2-s − 0.577·3-s − 0.113·4-s − 1.48·5-s − 0.543·6-s − 0.579·7-s − 1.04·8-s + 0.333·9-s − 1.39·10-s − 0.301·11-s + 0.0653·12-s − 0.546·14-s + 0.855·15-s − 0.874·16-s + 1.44·17-s + 0.313·18-s + 1.48·19-s + 0.167·20-s + 0.334·21-s − 0.283·22-s + 1.10·23-s + 0.605·24-s + 1.19·25-s − 0.192·27-s + 0.0656·28-s − 0.0355·29-s + 0.805·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.33T + 2T^{2} \) |
| 5 | \( 1 + 3.31T + 5T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 17 | \( 1 - 5.96T + 17T^{2} \) |
| 19 | \( 1 - 6.46T + 19T^{2} \) |
| 23 | \( 1 - 5.32T + 23T^{2} \) |
| 29 | \( 1 + 0.191T + 29T^{2} \) |
| 31 | \( 1 + 9.38T + 31T^{2} \) |
| 37 | \( 1 - 9.26T + 37T^{2} \) |
| 41 | \( 1 - 0.250T + 41T^{2} \) |
| 43 | \( 1 + 5.16T + 43T^{2} \) |
| 47 | \( 1 + 4.36T + 47T^{2} \) |
| 53 | \( 1 - 7.13T + 53T^{2} \) |
| 59 | \( 1 + 9.80T + 59T^{2} \) |
| 61 | \( 1 - 8.46T + 61T^{2} \) |
| 67 | \( 1 + 6.81T + 67T^{2} \) |
| 71 | \( 1 + 7.41T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 6.63T + 79T^{2} \) |
| 83 | \( 1 + 9.84T + 83T^{2} \) |
| 89 | \( 1 - 5.37T + 89T^{2} \) |
| 97 | \( 1 + 7.24T + 97T^{2} \) |
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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
−7.57137166454170156318025239962, −7.13539048197596280080617894543, −6.15564889249007486997685310430, −5.39157893735887924071960477477, −4.95800901347098149843293190699, −4.04835640526570916764434614723, −3.40035198354971759339834055681, −2.98200943124251675413052613351, −1.03864275584036559441033612239, 0,
1.03864275584036559441033612239, 2.98200943124251675413052613351, 3.40035198354971759339834055681, 4.04835640526570916764434614723, 4.95800901347098149843293190699, 5.39157893735887924071960477477, 6.15564889249007486997685310430, 7.13539048197596280080617894543, 7.57137166454170156318025239962