| L(s) = 1 | − 2.90·3-s − 3.52·5-s + 3.26·7-s + 5.43·9-s − 4.64·11-s − 0.0450·13-s + 10.2·15-s − 5.09·17-s + 2.40·19-s − 9.49·21-s − 1.86·23-s + 7.42·25-s − 7.08·27-s − 1.84·29-s − 3.76·31-s + 13.5·33-s − 11.5·35-s + 8.39·37-s + 0.130·39-s + 3.06·41-s − 7.51·43-s − 19.1·45-s + 8.33·47-s + 3.68·49-s + 14.8·51-s − 10.2·53-s + 16.3·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s − 1.57·5-s + 1.23·7-s + 1.81·9-s − 1.40·11-s − 0.0124·13-s + 2.64·15-s − 1.23·17-s + 0.552·19-s − 2.07·21-s − 0.388·23-s + 1.48·25-s − 1.36·27-s − 0.342·29-s − 0.676·31-s + 2.35·33-s − 1.94·35-s + 1.38·37-s + 0.0209·39-s + 0.478·41-s − 1.14·43-s − 2.85·45-s + 1.21·47-s + 0.526·49-s + 2.07·51-s − 1.41·53-s + 2.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2861664783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2861664783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 5 | \( 1 + 3.52T + 5T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 11 | \( 1 + 4.64T + 11T^{2} \) |
| 13 | \( 1 + 0.0450T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 - 2.40T + 19T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 + 1.84T + 29T^{2} \) |
| 31 | \( 1 + 3.76T + 31T^{2} \) |
| 37 | \( 1 - 8.39T + 37T^{2} \) |
| 41 | \( 1 - 3.06T + 41T^{2} \) |
| 43 | \( 1 + 7.51T + 43T^{2} \) |
| 47 | \( 1 - 8.33T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 2.35T + 59T^{2} \) |
| 61 | \( 1 + 3.15T + 61T^{2} \) |
| 67 | \( 1 + 6.89T + 67T^{2} \) |
| 71 | \( 1 + 0.257T + 71T^{2} \) |
| 73 | \( 1 + 7.34T + 73T^{2} \) |
| 79 | \( 1 + 8.04T + 79T^{2} \) |
| 83 | \( 1 - 4.16T + 83T^{2} \) |
| 89 | \( 1 + 4.10T + 89T^{2} \) |
| 97 | \( 1 + 7.27T + 97T^{2} \) |
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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
−7.81322490652885855352862223573, −7.49686641588341733181994325453, −6.71351987535625495642693315598, −5.76190717037849111546587214045, −5.15171970578953643863819862611, −4.52423021105029763977132495281, −4.12073510476332228253806884409, −2.80245585502110579099999223736, −1.52196596554571371099449574015, −0.31552112041097980567464334824,
0.31552112041097980567464334824, 1.52196596554571371099449574015, 2.80245585502110579099999223736, 4.12073510476332228253806884409, 4.52423021105029763977132495281, 5.15171970578953643863819862611, 5.76190717037849111546587214045, 6.71351987535625495642693315598, 7.49686641588341733181994325453, 7.81322490652885855352862223573
