| L(s) = 1 | + 7·4-s + 48·11-s − 15·16-s + 248·19-s − 156·29-s + 400·31-s − 660·41-s + 336·44-s + 286·49-s + 48·59-s − 644·61-s − 553·64-s + 576·71-s + 1.73e3·76-s + 1.04e3·79-s + 2.05e3·89-s + 3.46e3·101-s + 2.94e3·109-s − 1.09e3·116-s − 934·121-s + 2.80e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 7/8·4-s + 1.31·11-s − 0.234·16-s + 2.99·19-s − 0.998·29-s + 2.31·31-s − 2.51·41-s + 1.15·44-s + 0.833·49-s + 0.105·59-s − 1.35·61-s − 1.08·64-s + 0.962·71-s + 2.62·76-s + 1.48·79-s + 2.44·89-s + 3.41·101-s + 2.59·109-s − 0.874·116-s − 0.701·121-s + 2.02·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.772326996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.772326996\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 286 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 24 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1082 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6910 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 124 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 78 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 200 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 96406 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 330 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 150550 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 207070 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 95254 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 24 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 322 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 563110 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 288 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 593134 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 520 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1119238 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1026 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1743550 T^{2} + p^{6} T^{4} \) |
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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
−11.85742375934712410248591602161, −11.71294642365021137045619307040, −11.33642630778844182910884582444, −10.58739758967208568719861052139, −10.02169064585842822494792143390, −9.739406877349121147928622297093, −9.008561635846258010045545578359, −8.824369208884537885155669905303, −7.81338890642528832510599663318, −7.53668887876042297180634959305, −7.01991737506216189851429129492, −6.27192500406862216878999328735, −6.22973609684150742025376516686, −5.05541657464192386321649947905, −4.91037297096052416223215233555, −3.67838368001064953038658664428, −3.37114405990384096886681688832, −2.50061466087630483910122054536, −1.56912799836096494179487774155, −0.870406853240718195167808442241,
0.870406853240718195167808442241, 1.56912799836096494179487774155, 2.50061466087630483910122054536, 3.37114405990384096886681688832, 3.67838368001064953038658664428, 4.91037297096052416223215233555, 5.05541657464192386321649947905, 6.22973609684150742025376516686, 6.27192500406862216878999328735, 7.01991737506216189851429129492, 7.53668887876042297180634959305, 7.81338890642528832510599663318, 8.824369208884537885155669905303, 9.008561635846258010045545578359, 9.739406877349121147928622297093, 10.02169064585842822494792143390, 10.58739758967208568719861052139, 11.33642630778844182910884582444, 11.71294642365021137045619307040, 11.85742375934712410248591602161
