L(s) = 1 | + 1.20·3-s − 1.79·5-s + 2.78·7-s − 1.54·9-s − 0.183·11-s − 4.03·13-s − 2.17·15-s + 1.33·17-s + 0.976·19-s + 3.36·21-s + 6.76·23-s − 1.76·25-s − 5.48·27-s − 3.70·29-s − 5.56·31-s − 0.221·33-s − 5.01·35-s + 2.65·37-s − 4.87·39-s − 10.1·41-s + 2.03·43-s + 2.78·45-s − 11.3·47-s + 0.768·49-s + 1.60·51-s + 12.2·53-s + 0.330·55-s + ⋯ |
L(s) = 1 | + 0.696·3-s − 0.804·5-s + 1.05·7-s − 0.515·9-s − 0.0553·11-s − 1.12·13-s − 0.560·15-s + 0.322·17-s + 0.224·19-s + 0.733·21-s + 1.40·23-s − 0.352·25-s − 1.05·27-s − 0.687·29-s − 0.998·31-s − 0.0385·33-s − 0.847·35-s + 0.436·37-s − 0.780·39-s − 1.58·41-s + 0.309·43-s + 0.414·45-s − 1.65·47-s + 0.109·49-s + 0.224·51-s + 1.68·53-s + 0.0445·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 - 1.20T + 3T^{2} \) |
| 5 | \( 1 + 1.79T + 5T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 11 | \( 1 + 0.183T + 11T^{2} \) |
| 13 | \( 1 + 4.03T + 13T^{2} \) |
| 17 | \( 1 - 1.33T + 17T^{2} \) |
| 19 | \( 1 - 0.976T + 19T^{2} \) |
| 23 | \( 1 - 6.76T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 - 2.65T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 2.03T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.65T + 61T^{2} \) |
| 67 | \( 1 + 1.94T + 67T^{2} \) |
| 71 | \( 1 + 5.71T + 71T^{2} \) |
| 73 | \( 1 + 8.31T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 8.57T + 83T^{2} \) |
| 89 | \( 1 - 0.496T + 89T^{2} \) |
| 97 | \( 1 + 0.552T + 97T^{2} \) |
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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
−8.146596073907495017903494662463, −7.43133219221860934402988519536, −7.01496275981868554209835656012, −5.57841781776413019463792029615, −5.10015815685618256470235323303, −4.19242243579576732770840175382, −3.36270147215778036426060246630, −2.56410964652605455620336454075, −1.55952005505326650046201567769, 0,
1.55952005505326650046201567769, 2.56410964652605455620336454075, 3.36270147215778036426060246630, 4.19242243579576732770840175382, 5.10015815685618256470235323303, 5.57841781776413019463792029615, 7.01496275981868554209835656012, 7.43133219221860934402988519536, 8.146596073907495017903494662463