| L(s) = 1 | + 3.04·3-s − 5-s − 7-s + 6.29·9-s + 2.68·11-s + 13-s − 3.04·15-s + 3.20·17-s + 1.48·19-s − 3.04·21-s − 0.688·23-s + 25-s + 10.0·27-s + 0.361·29-s + 2.07·31-s + 8.19·33-s + 35-s − 9.88·37-s + 3.04·39-s − 1.24·41-s + 8.78·43-s − 6.29·45-s − 3.12·47-s + 49-s + 9.75·51-s + 7.12·53-s − 2.68·55-s + ⋯ |
| L(s) = 1 | + 1.76·3-s − 0.447·5-s − 0.377·7-s + 2.09·9-s + 0.810·11-s + 0.277·13-s − 0.787·15-s + 0.776·17-s + 0.341·19-s − 0.665·21-s − 0.143·23-s + 0.200·25-s + 1.93·27-s + 0.0670·29-s + 0.372·31-s + 1.42·33-s + 0.169·35-s − 1.62·37-s + 0.488·39-s − 0.195·41-s + 1.34·43-s − 0.939·45-s − 0.455·47-s + 0.142·49-s + 1.36·51-s + 0.978·53-s − 0.362·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.689738683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.689738683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 3.04T + 3T^{2} \) |
| 11 | \( 1 - 2.68T + 11T^{2} \) |
| 17 | \( 1 - 3.20T + 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 + 0.688T + 23T^{2} \) |
| 29 | \( 1 - 0.361T + 29T^{2} \) |
| 31 | \( 1 - 2.07T + 31T^{2} \) |
| 37 | \( 1 + 9.88T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 - 8.78T + 43T^{2} \) |
| 47 | \( 1 + 3.12T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 + 6.52T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 - 1.10T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 1.17T + 83T^{2} \) |
| 89 | \( 1 + 6.91T + 89T^{2} \) |
| 97 | \( 1 - 9.57T + 97T^{2} \) |
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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
−8.536351911718063350254198948879, −7.960671087013566617000634772260, −7.21801688742513989089726006859, −6.64221022844351712542860924844, −5.49716228762688941833075681116, −4.34297555559043945540694254356, −3.65512013985341843805823490562, −3.17158449475702291384005354620, −2.17368016616571420951213944137, −1.11444452364011462504642501112,
1.11444452364011462504642501112, 2.17368016616571420951213944137, 3.17158449475702291384005354620, 3.65512013985341843805823490562, 4.34297555559043945540694254356, 5.49716228762688941833075681116, 6.64221022844351712542860924844, 7.21801688742513989089726006859, 7.960671087013566617000634772260, 8.536351911718063350254198948879
