L(s) = 1 | + 2.66·3-s + 3.83·5-s + 0.592·7-s + 4.08·9-s + 2.69·11-s − 0.914·13-s + 10.1·15-s + 4.61·17-s − 4.04·19-s + 1.57·21-s + 1.00·23-s + 9.66·25-s + 2.87·27-s − 0.102·29-s − 5.16·31-s + 7.16·33-s + 2.26·35-s + 1.33·37-s − 2.43·39-s − 8.01·41-s − 0.552·43-s + 15.6·45-s + 4.45·47-s − 6.64·49-s + 12.2·51-s − 4.69·53-s + 10.3·55-s + ⋯ |
L(s) = 1 | + 1.53·3-s + 1.71·5-s + 0.223·7-s + 1.36·9-s + 0.811·11-s − 0.253·13-s + 2.63·15-s + 1.11·17-s − 0.928·19-s + 0.344·21-s + 0.209·23-s + 1.93·25-s + 0.553·27-s − 0.0190·29-s − 0.926·31-s + 1.24·33-s + 0.383·35-s + 0.219·37-s − 0.389·39-s − 1.25·41-s − 0.0842·43-s + 2.33·45-s + 0.650·47-s − 0.949·49-s + 1.72·51-s − 0.644·53-s + 1.39·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.103340828\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.103340828\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 - 2.66T + 3T^{2} \) |
| 5 | \( 1 - 3.83T + 5T^{2} \) |
| 7 | \( 1 - 0.592T + 7T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 + 0.914T + 13T^{2} \) |
| 17 | \( 1 - 4.61T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 - 1.00T + 23T^{2} \) |
| 29 | \( 1 + 0.102T + 29T^{2} \) |
| 31 | \( 1 + 5.16T + 31T^{2} \) |
| 37 | \( 1 - 1.33T + 37T^{2} \) |
| 41 | \( 1 + 8.01T + 41T^{2} \) |
| 43 | \( 1 + 0.552T + 43T^{2} \) |
| 47 | \( 1 - 4.45T + 47T^{2} \) |
| 53 | \( 1 + 4.69T + 53T^{2} \) |
| 59 | \( 1 + 7.37T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 + 8.16T + 67T^{2} \) |
| 71 | \( 1 + 7.48T + 71T^{2} \) |
| 73 | \( 1 - 7.15T + 73T^{2} \) |
| 79 | \( 1 - 1.67T + 79T^{2} \) |
| 83 | \( 1 + 1.36T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.647129967616994041952287548741, −7.85417200217601678962115046760, −7.03500619439244947997337684593, −6.27763540088032400803751510169, −5.52593167094951851168426286613, −4.61890171898097387694681658619, −3.58837342715775426153090758619, −2.85192991759583626566753682028, −1.95963746535171816104512498311, −1.45492879689212154191929150668,
1.45492879689212154191929150668, 1.95963746535171816104512498311, 2.85192991759583626566753682028, 3.58837342715775426153090758619, 4.61890171898097387694681658619, 5.52593167094951851168426286613, 6.27763540088032400803751510169, 7.03500619439244947997337684593, 7.85417200217601678962115046760, 8.647129967616994041952287548741