| L(s) = 1 | + 32·7-s − 96·25-s − 160·31-s + 444·49-s − 384·73-s + 416·79-s − 320·97-s − 288·103-s − 228·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 548·169-s + 173-s − 3.07e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 32/7·7-s − 3.83·25-s − 5.16·31-s + 9.06·49-s − 5.26·73-s + 5.26·79-s − 3.29·97-s − 2.79·103-s − 1.88·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.24·169-s + 0.00578·173-s − 17.5·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01023455266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01023455266\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 48 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 114 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 274 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 416 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 302 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 240 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2062 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 1056 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3442 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4290 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 4560 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 6450 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 4526 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 2578 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 3810 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 3410 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 9792 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.84999607375970083617114219141, −5.80984837200876417176251954647, −5.75943644100450702170109709331, −5.55390471695389189779737610933, −5.38736126733883013672116131371, −5.24078093828104255793175415836, −4.97470192043978893374893742831, −4.62574532944891604562867131102, −4.59114699715124120187282819571, −4.35916931871173015658490982558, −4.13201178127763089585665144340, −3.92405953509804139886141578797, −3.70223185631707715247368932231, −3.42829475589692927911913334673, −3.35517988414858821496958386321, −2.72369529685937526103488588961, −2.41834308581735007416793662188, −2.16242083824564045951643223077, −1.92919329826591436508112531798, −1.72671239266931937375183284990, −1.65981445682039772320372174541, −1.42679901976412576481370252411, −1.25945247615183311690766855539, −0.52731197680205007433539628504, −0.01068664892415829888948771175,
0.01068664892415829888948771175, 0.52731197680205007433539628504, 1.25945247615183311690766855539, 1.42679901976412576481370252411, 1.65981445682039772320372174541, 1.72671239266931937375183284990, 1.92919329826591436508112531798, 2.16242083824564045951643223077, 2.41834308581735007416793662188, 2.72369529685937526103488588961, 3.35517988414858821496958386321, 3.42829475589692927911913334673, 3.70223185631707715247368932231, 3.92405953509804139886141578797, 4.13201178127763089585665144340, 4.35916931871173015658490982558, 4.59114699715124120187282819571, 4.62574532944891604562867131102, 4.97470192043978893374893742831, 5.24078093828104255793175415836, 5.38736126733883013672116131371, 5.55390471695389189779737610933, 5.75943644100450702170109709331, 5.80984837200876417176251954647, 5.84999607375970083617114219141
