L(s) = 1 | − 0.893·3-s − 3.85·5-s + 3.77·7-s − 2.20·9-s − 4.71·11-s − 0.511·13-s + 3.44·15-s + 5.63·17-s + 0.0695·19-s − 3.37·21-s + 5.20·23-s + 9.89·25-s + 4.64·27-s − 1.66·29-s − 0.370·31-s + 4.21·33-s − 14.5·35-s + 0.932·37-s + 0.457·39-s + 5.06·41-s − 6.38·43-s + 8.49·45-s − 4.19·47-s + 7.26·49-s − 5.03·51-s + 1.31·53-s + 18.2·55-s + ⋯ |
L(s) = 1 | − 0.516·3-s − 1.72·5-s + 1.42·7-s − 0.733·9-s − 1.42·11-s − 0.141·13-s + 0.890·15-s + 1.36·17-s + 0.0159·19-s − 0.736·21-s + 1.08·23-s + 1.97·25-s + 0.894·27-s − 0.310·29-s − 0.0665·31-s + 0.734·33-s − 2.46·35-s + 0.153·37-s + 0.0732·39-s + 0.790·41-s − 0.974·43-s + 1.26·45-s − 0.612·47-s + 1.03·49-s − 0.704·51-s + 0.179·53-s + 2.45·55-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + 0.893T + 3T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 - 3.77T + 7T^{2} \) |
| 11 | \( 1 + 4.71T + 11T^{2} \) |
| 13 | \( 1 + 0.511T + 13T^{2} \) |
| 17 | \( 1 - 5.63T + 17T^{2} \) |
| 19 | \( 1 - 0.0695T + 19T^{2} \) |
| 23 | \( 1 - 5.20T + 23T^{2} \) |
| 29 | \( 1 + 1.66T + 29T^{2} \) |
| 31 | \( 1 + 0.370T + 31T^{2} \) |
| 37 | \( 1 - 0.932T + 37T^{2} \) |
| 41 | \( 1 - 5.06T + 41T^{2} \) |
| 43 | \( 1 + 6.38T + 43T^{2} \) |
| 47 | \( 1 + 4.19T + 47T^{2} \) |
| 53 | \( 1 - 1.31T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 2.09T + 67T^{2} \) |
| 71 | \( 1 - 3.30T + 71T^{2} \) |
| 73 | \( 1 - 1.07T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−8.126609733145360634147637842247, −7.60524660815463278331290624693, −6.80743709779180425779459386692, −5.43514615487249139194173789930, −5.18614621298227000797854505842, −4.42775339422866905708810640372, −3.41539721637682870904107584499, −2.65302055557902992208135149418, −1.11718045848856775206684795486, 0,
1.11718045848856775206684795486, 2.65302055557902992208135149418, 3.41539721637682870904107584499, 4.42775339422866905708810640372, 5.18614621298227000797854505842, 5.43514615487249139194173789930, 6.80743709779180425779459386692, 7.60524660815463278331290624693, 8.126609733145360634147637842247