| L(s) = 1 | + 2·2-s + 3·4-s − 3·5-s + 2·7-s + 4·8-s − 6·10-s − 3·11-s + 4·13-s + 4·14-s + 5·16-s + 3·17-s + 10·19-s − 9·20-s − 6·22-s + 9·23-s + 5·25-s + 8·26-s + 6·28-s + 6·29-s + 4·31-s + 6·32-s + 6·34-s − 6·35-s + 4·37-s + 20·38-s − 12·40-s − 15·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.34·5-s + 0.755·7-s + 1.41·8-s − 1.89·10-s − 0.904·11-s + 1.10·13-s + 1.06·14-s + 5/4·16-s + 0.727·17-s + 2.29·19-s − 2.01·20-s − 1.27·22-s + 1.87·23-s + 25-s + 1.56·26-s + 1.13·28-s + 1.11·29-s + 0.718·31-s + 1.06·32-s + 1.02·34-s − 1.01·35-s + 0.657·37-s + 3.24·38-s − 1.89·40-s − 2.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.888057301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.888057301\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 15 T + 130 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 96 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 120 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−9.989311466706173968858372992494, −9.954520638479619936727954260948, −9.076260592322623027474928498255, −8.572929098740636097092393616892, −8.228202998927323849821564884408, −7.907782647672812351449838655488, −7.41378724699717789879241931549, −7.19215216282486385928762664991, −6.68189466474888002259540400955, −6.20584167743436367317175733007, −5.41101142589322471258200350634, −5.28466611498300721165837773302, −4.91072149991145697299504003910, −4.46922305979512963531133136284, −3.74336597601820203487671785825, −3.50488156228185864364923133175, −2.91612008550503675201894902122, −2.66461597644958671342222143631, −1.34713470932428067038240220037, −1.01413063779917027532365556795,
1.01413063779917027532365556795, 1.34713470932428067038240220037, 2.66461597644958671342222143631, 2.91612008550503675201894902122, 3.50488156228185864364923133175, 3.74336597601820203487671785825, 4.46922305979512963531133136284, 4.91072149991145697299504003910, 5.28466611498300721165837773302, 5.41101142589322471258200350634, 6.20584167743436367317175733007, 6.68189466474888002259540400955, 7.19215216282486385928762664991, 7.41378724699717789879241931549, 7.907782647672812351449838655488, 8.228202998927323849821564884408, 8.572929098740636097092393616892, 9.076260592322623027474928498255, 9.954520638479619936727954260948, 9.989311466706173968858372992494
