| L(s) = 1 | − 2.77·2-s + 5.71·4-s + 5-s − 3.62·7-s − 10.3·8-s − 2.77·10-s − 3.37·11-s + 10.0·14-s + 17.2·16-s − 6.65·17-s − 0.620·19-s + 5.71·20-s + 9.38·22-s + 1.72·23-s + 25-s − 20.7·28-s + 6.09·29-s + 2.62·31-s − 27.3·32-s + 18.4·34-s − 3.62·35-s + 7.72·37-s + 1.72·38-s − 10.3·40-s + 3.10·41-s − 3.82·43-s − 19.3·44-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 2.85·4-s + 0.447·5-s − 1.37·7-s − 3.65·8-s − 0.878·10-s − 1.01·11-s + 2.69·14-s + 4.31·16-s − 1.61·17-s − 0.142·19-s + 1.27·20-s + 2.00·22-s + 0.359·23-s + 0.200·25-s − 3.91·28-s + 1.13·29-s + 0.472·31-s − 4.82·32-s + 3.16·34-s − 0.612·35-s + 1.26·37-s + 0.279·38-s − 1.63·40-s + 0.484·41-s − 0.583·43-s − 2.91·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 7 | \( 1 + 3.62T + 7T^{2} \) |
| 11 | \( 1 + 3.37T + 11T^{2} \) |
| 17 | \( 1 + 6.65T + 17T^{2} \) |
| 19 | \( 1 + 0.620T + 19T^{2} \) |
| 23 | \( 1 - 1.72T + 23T^{2} \) |
| 29 | \( 1 - 6.09T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 - 7.72T + 37T^{2} \) |
| 41 | \( 1 - 3.10T + 41T^{2} \) |
| 43 | \( 1 + 3.82T + 43T^{2} \) |
| 47 | \( 1 + 5.00T + 47T^{2} \) |
| 53 | \( 1 - 3.97T + 53T^{2} \) |
| 59 | \( 1 + 3.54T + 59T^{2} \) |
| 61 | \( 1 - 2.01T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 6.52T + 71T^{2} \) |
| 73 | \( 1 - 6.23T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 4.33T + 83T^{2} \) |
| 89 | \( 1 - 8.82T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.75752956703285928216546016814, −6.78184883846372691322210820696, −6.54845816307656359702151554241, −5.94760539510262077291559570021, −4.85925463778202141446657509163, −3.45511768199529560443485682789, −2.59063335416827647153940551567, −2.26594172784104123106014114764, −0.898014913495587010420870700838, 0,
0.898014913495587010420870700838, 2.26594172784104123106014114764, 2.59063335416827647153940551567, 3.45511768199529560443485682789, 4.85925463778202141446657509163, 5.94760539510262077291559570021, 6.54845816307656359702151554241, 6.78184883846372691322210820696, 7.75752956703285928216546016814
