| L(s) = 1 | − 3-s − 3·5-s + 9-s − 6·11-s − 13-s + 3·15-s − 17-s + 4·19-s + 4·23-s + 4·25-s − 27-s + 7·29-s + 7·31-s + 6·33-s − 8·37-s + 39-s + 3·41-s + 8·43-s − 3·45-s + 7·47-s + 51-s + 4·53-s + 18·55-s − 4·57-s + 5·59-s − 4·61-s + 3·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s − 1.80·11-s − 0.277·13-s + 0.774·15-s − 0.242·17-s + 0.917·19-s + 0.834·23-s + 4/5·25-s − 0.192·27-s + 1.29·29-s + 1.25·31-s + 1.04·33-s − 1.31·37-s + 0.160·39-s + 0.468·41-s + 1.21·43-s − 0.447·45-s + 1.02·47-s + 0.140·51-s + 0.549·53-s + 2.42·55-s − 0.529·57-s + 0.650·59-s − 0.512·61-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.43870524448973, −12.96041951523431, −12.38440243452137, −12.14470013716818, −11.61874181563780, −11.21051427107480, −10.67342547154118, −10.26031037036660, −9.977625000611043, −9.086315037873462, −8.576935769527345, −8.189582589233443, −7.540209916691800, −7.293012053070906, −6.927648031537190, −6.004850968664673, −5.613891718249308, −4.993173897713291, −4.577134254959298, −4.183354467828927, −3.319216204399272, −2.844488918102347, −2.430868344660166, −1.273026042773888, −0.6476624225379856, 0,
0.6476624225379856, 1.273026042773888, 2.430868344660166, 2.844488918102347, 3.319216204399272, 4.183354467828927, 4.577134254959298, 4.993173897713291, 5.613891718249308, 6.004850968664673, 6.927648031537190, 7.293012053070906, 7.540209916691800, 8.189582589233443, 8.576935769527345, 9.086315037873462, 9.977625000611043, 10.26031037036660, 10.67342547154118, 11.21051427107480, 11.61874181563780, 12.14470013716818, 12.38440243452137, 12.96041951523431, 13.43870524448973
