| L(s) = 1 | + 2·3-s + 96·5-s − 239·9-s − 720·11-s − 572·13-s + 192·15-s − 1.25e3·17-s + 94·19-s + 96·23-s + 6.09e3·25-s − 964·27-s − 4.37e3·29-s + 6.24e3·31-s − 1.44e3·33-s − 1.07e4·37-s − 1.14e3·39-s − 1.20e4·41-s − 9.16e3·43-s − 2.29e4·45-s + 2.58e4·47-s − 2.50e3·51-s + 1.01e3·53-s − 6.91e4·55-s + 188·57-s − 1.24e3·59-s − 7.59e3·61-s − 5.49e4·65-s + ⋯ |
| L(s) = 1 | + 0.128·3-s + 1.71·5-s − 0.983·9-s − 1.79·11-s − 0.938·13-s + 0.220·15-s − 1.05·17-s + 0.0597·19-s + 0.0378·23-s + 1.94·25-s − 0.254·27-s − 0.965·29-s + 1.16·31-s − 0.230·33-s − 1.29·37-s − 0.120·39-s − 1.11·41-s − 0.755·43-s − 1.68·45-s + 1.70·47-s − 0.135·51-s + 0.0495·53-s − 3.08·55-s + 0.00766·57-s − 0.0464·59-s − 0.261·61-s − 1.61·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 2 T + p^{5} T^{2} \) |
| 5 | \( 1 - 96 T + p^{5} T^{2} \) |
| 11 | \( 1 + 720 T + p^{5} T^{2} \) |
| 13 | \( 1 + 44 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 1254 T + p^{5} T^{2} \) |
| 19 | \( 1 - 94 T + p^{5} T^{2} \) |
| 23 | \( 1 - 96 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4374 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6244 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10798 T + p^{5} T^{2} \) |
| 41 | \( 1 + 12006 T + p^{5} T^{2} \) |
| 43 | \( 1 + 9160 T + p^{5} T^{2} \) |
| 47 | \( 1 - 25836 T + p^{5} T^{2} \) |
| 53 | \( 1 - 1014 T + p^{5} T^{2} \) |
| 59 | \( 1 + 1242 T + p^{5} T^{2} \) |
| 61 | \( 1 + 7592 T + p^{5} T^{2} \) |
| 67 | \( 1 - 41132 T + p^{5} T^{2} \) |
| 71 | \( 1 + 37632 T + p^{5} T^{2} \) |
| 73 | \( 1 - 13438 T + p^{5} T^{2} \) |
| 79 | \( 1 - 6248 T + p^{5} T^{2} \) |
| 83 | \( 1 - 25254 T + p^{5} T^{2} \) |
| 89 | \( 1 - 45126 T + p^{5} T^{2} \) |
| 97 | \( 1 + 107222 T + p^{5} T^{2} \) |
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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
−10.84152863973636487883247785398, −10.15605348473926950695426672529, −9.219316960045859883403882543355, −8.221602676997994464285168248086, −6.85585314601525906549664239080, −5.63789707889400918395397644706, −5.04176549946548992694004507212, −2.78598837853104579456271448294, −2.09961319601010981579420234783, 0,
2.09961319601010981579420234783, 2.78598837853104579456271448294, 5.04176549946548992694004507212, 5.63789707889400918395397644706, 6.85585314601525906549664239080, 8.221602676997994464285168248086, 9.219316960045859883403882543355, 10.15605348473926950695426672529, 10.84152863973636487883247785398
