L(s) = 1 | − 9-s + 2·11-s − 4·16-s + 2·19-s + 4·29-s − 12·31-s − 11·49-s + 16·59-s − 2·61-s − 24·71-s − 32·79-s + 81-s + 12·89-s − 2·99-s + 4·101-s − 8·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | − 1/3·9-s + 0.603·11-s − 16-s + 0.458·19-s + 0.742·29-s − 2.15·31-s − 1.57·49-s + 2.08·59-s − 0.256·61-s − 2.84·71-s − 3.60·79-s + 1/9·81-s + 1.27·89-s − 0.201·99-s + 0.398·101-s − 0.766·109-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2030625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2030625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.471995234\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.471995234\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.548766757501507698246424256944, −9.413536435927328250175960390265, −8.919360813747291069389790560515, −8.559541758158458095367011329787, −8.333437702861934967332757652589, −7.51918595249373066857123824590, −7.43488676905309528588313224795, −6.82464734474922463008327271495, −6.62360351443807592976756319500, −6.03442508510505748600502576477, −5.44590430097588137327209655755, −5.41081426384085138124804158955, −4.49112833241380922224768471874, −4.36002682787085093088138193913, −3.75895657612691884878267342316, −3.07564879122154319068832060598, −2.85061582980062512808896909837, −1.89387420170770766800380050930, −1.59333984936815195897336748818, −0.48458776575377232738719977766,
0.48458776575377232738719977766, 1.59333984936815195897336748818, 1.89387420170770766800380050930, 2.85061582980062512808896909837, 3.07564879122154319068832060598, 3.75895657612691884878267342316, 4.36002682787085093088138193913, 4.49112833241380922224768471874, 5.41081426384085138124804158955, 5.44590430097588137327209655755, 6.03442508510505748600502576477, 6.62360351443807592976756319500, 6.82464734474922463008327271495, 7.43488676905309528588313224795, 7.51918595249373066857123824590, 8.333437702861934967332757652589, 8.559541758158458095367011329787, 8.919360813747291069389790560515, 9.413536435927328250175960390265, 9.548766757501507698246424256944