| L(s) = 1 | + 7·4-s + 29·9-s + 120·11-s − 15·16-s + 122·19-s − 138·29-s − 62·31-s + 203·36-s − 12·41-s + 840·44-s + 202·49-s + 1.50e3·59-s + 70·61-s − 553·64-s − 198·71-s + 854·76-s − 976·79-s + 112·81-s − 2.43e3·89-s + 3.48e3·99-s + 900·101-s − 3.47e3·109-s − 966·116-s + 8.13e3·121-s − 434·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 7/8·4-s + 1.07·9-s + 3.28·11-s − 0.234·16-s + 1.47·19-s − 0.883·29-s − 0.359·31-s + 0.939·36-s − 0.0457·41-s + 2.87·44-s + 0.588·49-s + 3.32·59-s + 0.146·61-s − 1.08·64-s − 0.330·71-s + 1.28·76-s − 1.38·79-s + 0.153·81-s − 2.89·89-s + 3.53·99-s + 0.886·101-s − 3.05·109-s − 0.773·116-s + 6.11·121-s − 0.314·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.744752385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.744752385\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 29 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 202 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 60 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3433 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 61 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 18250 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 69 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 98170 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 130430 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8579 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 224975 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 753 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 35 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 497842 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 99 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 483095 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 488 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 417670 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1215 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1464145 T^{2} + p^{6} T^{4} \) |
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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
−11.21219505456552640037108400351, −10.54754195189136177944178651436, −9.994503303942729494075999201032, −9.482007864805653979468456546494, −9.391856245054443361365465584282, −8.816212109114210513782772180720, −8.323879995802266934735424163363, −7.48372947879836718651895072872, −7.11505561263418080937064605666, −6.74294886470314225379957839295, −6.64709698384608832953484020100, −5.73771771915643621506658024951, −5.44588159960337120073855391967, −4.33241890101119398733755401396, −4.10184364383640938665762479633, −3.63696255267406558449950485915, −2.86518839373075151352006329301, −1.85887681046350446068606899067, −1.45504075577426056810832806750, −0.874448559364318066784218900091,
0.874448559364318066784218900091, 1.45504075577426056810832806750, 1.85887681046350446068606899067, 2.86518839373075151352006329301, 3.63696255267406558449950485915, 4.10184364383640938665762479633, 4.33241890101119398733755401396, 5.44588159960337120073855391967, 5.73771771915643621506658024951, 6.64709698384608832953484020100, 6.74294886470314225379957839295, 7.11505561263418080937064605666, 7.48372947879836718651895072872, 8.323879995802266934735424163363, 8.816212109114210513782772180720, 9.391856245054443361365465584282, 9.482007864805653979468456546494, 9.994503303942729494075999201032, 10.54754195189136177944178651436, 11.21219505456552640037108400351
