| L(s) = 1 | − 2·4-s − 22·7-s + 14·13-s + 4·16-s + 58·19-s + 44·28-s + 58·31-s − 112·37-s − 10·43-s + 265·49-s − 28·52-s − 110·61-s − 8·64-s + 74·67-s + 32·73-s − 116·76-s + 208·79-s − 308·91-s − 82·97-s − 124·103-s − 338·109-s − 88·112-s + 224·121-s − 116·124-s + 127-s + 131-s − 1.27e3·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3.14·7-s + 1.07·13-s + 1/4·16-s + 3.05·19-s + 11/7·28-s + 1.87·31-s − 3.02·37-s − 0.232·43-s + 5.40·49-s − 0.538·52-s − 1.80·61-s − 1/8·64-s + 1.10·67-s + 0.438·73-s − 1.52·76-s + 2.63·79-s − 3.38·91-s − 0.845·97-s − 1.20·103-s − 3.10·109-s − 0.785·112-s + 1.85·121-s − 0.935·124-s + 0.00787·127-s + 0.00763·131-s − 9.59·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.018334731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.018334731\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 224 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 416 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 29 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 400 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 496 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 29 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 368 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1010 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6080 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8930 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12896 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2590 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 41 T + p^{2} T^{2} )^{2} \) |
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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
−10.93750925732094331592233578922, −10.50806708352987073728463607769, −10.01728386729680325309610404538, −9.714894404055168793415606505936, −9.414546684122472259610218473176, −9.079208675113970010263519147159, −8.506250090706393984644227828349, −7.940404867515878346551028690259, −7.24812666845456991924989407518, −6.89784285057407840103584957874, −6.30244831847803846395034121497, −6.20057023273435171462615140870, −5.23660426443313779508438002885, −5.20643914126522946995186713456, −3.92424570750486376887123028116, −3.56846273597023986863147741493, −3.16160740241260284587211403483, −2.79452488949178472009365443193, −1.28379398036722079984577032314, −0.46944165228859422603865026333,
0.46944165228859422603865026333, 1.28379398036722079984577032314, 2.79452488949178472009365443193, 3.16160740241260284587211403483, 3.56846273597023986863147741493, 3.92424570750486376887123028116, 5.20643914126522946995186713456, 5.23660426443313779508438002885, 6.20057023273435171462615140870, 6.30244831847803846395034121497, 6.89784285057407840103584957874, 7.24812666845456991924989407518, 7.940404867515878346551028690259, 8.506250090706393984644227828349, 9.079208675113970010263519147159, 9.414546684122472259610218473176, 9.714894404055168793415606505936, 10.01728386729680325309610404538, 10.50806708352987073728463607769, 10.93750925732094331592233578922
