| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 2·9-s − 2·11-s + 5·16-s + 4·18-s − 4·22-s + 12·23-s − 10·25-s − 8·29-s + 6·32-s + 6·36-s + 4·37-s + 20·43-s − 6·44-s + 24·46-s − 20·50-s + 4·53-s − 16·58-s + 7·64-s + 16·67-s + 32·71-s + 8·72-s + 8·74-s − 16·79-s − 5·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 2/3·9-s − 0.603·11-s + 5/4·16-s + 0.942·18-s − 0.852·22-s + 2.50·23-s − 2·25-s − 1.48·29-s + 1.06·32-s + 36-s + 0.657·37-s + 3.04·43-s − 0.904·44-s + 3.53·46-s − 2.82·50-s + 0.549·53-s − 2.10·58-s + 7/8·64-s + 1.95·67-s + 3.79·71-s + 0.942·72-s + 0.929·74-s − 1.80·79-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.879845969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.879845969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.11042783006774044806973272608, −9.664468343642505342977517911174, −9.416757680657536506691178342452, −8.942429623432213009222951527686, −8.274134519278857735927129108357, −7.76566383835216614641179076709, −7.47378900406172545797151789958, −7.18888155751611006972057629735, −6.60264138382635230738572977583, −6.27318266217535869909217754411, −5.54323956136095152735740092847, −5.38617944588913588926723289746, −5.04620414196187682688602921525, −4.24751455046883416199796491108, −3.95091820987185959332967484972, −3.65613553033659939715923863264, −2.66257201384197164240263590851, −2.56618247751545306602543247206, −1.74212596595951751211122154111, −0.890438696955777795245481394781,
0.890438696955777795245481394781, 1.74212596595951751211122154111, 2.56618247751545306602543247206, 2.66257201384197164240263590851, 3.65613553033659939715923863264, 3.95091820987185959332967484972, 4.24751455046883416199796491108, 5.04620414196187682688602921525, 5.38617944588913588926723289746, 5.54323956136095152735740092847, 6.27318266217535869909217754411, 6.60264138382635230738572977583, 7.18888155751611006972057629735, 7.47378900406172545797151789958, 7.76566383835216614641179076709, 8.274134519278857735927129108357, 8.942429623432213009222951527686, 9.416757680657536506691178342452, 9.664468343642505342977517911174, 10.11042783006774044806973272608
