| L(s) = 1 | + 3.86·5-s + 1.03·7-s + 5.46·11-s + 13-s + 5.65·17-s − 6.69·19-s + 6.92·23-s + 9.92·25-s + 7.72·29-s + 1.03·31-s + 3.99·35-s + 2·37-s + 3.86·41-s − 5.65·43-s − 9.46·47-s − 5.92·49-s − 5.65·53-s + 21.1·55-s + 2.53·59-s − 4.92·61-s + 3.86·65-s + 6.69·67-s − 9.46·71-s − 15.8·73-s + 5.65·77-s + 13.4·83-s + 21.8·85-s + ⋯ |
| L(s) = 1 | + 1.72·5-s + 0.391·7-s + 1.64·11-s + 0.277·13-s + 1.37·17-s − 1.53·19-s + 1.44·23-s + 1.98·25-s + 1.43·29-s + 0.185·31-s + 0.676·35-s + 0.328·37-s + 0.603·41-s − 0.862·43-s − 1.38·47-s − 0.846·49-s − 0.777·53-s + 2.84·55-s + 0.330·59-s − 0.630·61-s + 0.479·65-s + 0.817·67-s − 1.12·71-s − 1.85·73-s + 0.644·77-s + 1.47·83-s + 2.37·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.963609990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.963609990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.86T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 6.69T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 7.72T + 29T^{2} \) |
| 31 | \( 1 - 1.03T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 3.86T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 6.69T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 0.277T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.029397135283506686642927896268, −6.87847752594045058943014247630, −6.41730052715339418894200475403, −5.99545783443330840916134080907, −5.06549144083733197395817664192, −4.51784935307277644571954731744, −3.42784873139023080915421273517, −2.62667174453577556563709017325, −1.54993053227019667782288358632, −1.20061488155297657311213443464,
1.20061488155297657311213443464, 1.54993053227019667782288358632, 2.62667174453577556563709017325, 3.42784873139023080915421273517, 4.51784935307277644571954731744, 5.06549144083733197395817664192, 5.99545783443330840916134080907, 6.41730052715339418894200475403, 6.87847752594045058943014247630, 8.029397135283506686642927896268
