| L(s) = 1 | − 0.261·3-s − 19.5·5-s − 26.9·9-s + 56.2·11-s + 66.0·13-s + 5.10·15-s + 18.8·17-s − 80.8·19-s + 89.3·23-s + 255.·25-s + 14.1·27-s + 104.·29-s − 148.·31-s − 14.7·33-s + 13.8·37-s − 17.2·39-s − 174.·41-s − 205.·43-s + 525.·45-s − 116.·47-s − 4.92·51-s − 316.·53-s − 1.09e3·55-s + 21.1·57-s − 539.·59-s − 145.·61-s − 1.28e3·65-s + ⋯ |
| L(s) = 1 | − 0.0503·3-s − 1.74·5-s − 0.997·9-s + 1.54·11-s + 1.40·13-s + 0.0877·15-s + 0.268·17-s − 0.975·19-s + 0.810·23-s + 2.04·25-s + 0.100·27-s + 0.666·29-s − 0.860·31-s − 0.0776·33-s + 0.0613·37-s − 0.0709·39-s − 0.664·41-s − 0.728·43-s + 1.73·45-s − 0.361·47-s − 0.0135·51-s − 0.819·53-s − 2.69·55-s + 0.0491·57-s − 1.19·59-s − 0.305·61-s − 2.45·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.261T + 27T^{2} \) |
| 5 | \( 1 + 19.5T + 125T^{2} \) |
| 11 | \( 1 - 56.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 18.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 89.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 104.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 13.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 174.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 205.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 116.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 316.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 539.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 145.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 476.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 131.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 349.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 806.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 233.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 535.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 80.7T + 9.12e5T^{2} \) |
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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
−9.094531888579128151641614157960, −8.626292833038692566204701147056, −7.925261679616606465171098858084, −6.81203180712960263712962753704, −6.14089927688818026404094170134, −4.73425462129257658373556318860, −3.79293227684703683424891281121, −3.22045440687378144359677230558, −1.27038967029395030757052584668, 0,
1.27038967029395030757052584668, 3.22045440687378144359677230558, 3.79293227684703683424891281121, 4.73425462129257658373556318860, 6.14089927688818026404094170134, 6.81203180712960263712962753704, 7.925261679616606465171098858084, 8.626292833038692566204701147056, 9.094531888579128151641614157960
