L(s) = 1 | + 6·4-s − 7·9-s − 5·16-s − 28·19-s + 168·31-s − 42·36-s + 196·49-s − 32·61-s − 180·64-s − 168·76-s − 48·79-s − 32·81-s + 352·109-s − 66·121-s + 1.00e3·124-s + 127-s + 131-s + 137-s + 139-s + 35·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 476·169-s + 196·171-s + ⋯ |
L(s) = 1 | + 3/2·4-s − 7/9·9-s − 0.312·16-s − 1.47·19-s + 5.41·31-s − 7/6·36-s + 4·49-s − 0.524·61-s − 2.81·64-s − 2.21·76-s − 0.607·79-s − 0.395·81-s + 3.22·109-s − 0.545·121-s + 8.12·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.243·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.81·169-s + 1.14·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.301921419\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.301921419\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2^2$ | \( 1 + 7 T^{2} + p^{4} T^{4} \) |
| 5 | | \( 1 \) |
good | 2 | $C_2^2$ | \( ( 1 - 3 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 p T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 567 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 662 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 582 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 1138 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3087 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 1198 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2262 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 3462 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 2562 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 6953 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 8982 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 9433 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 8927 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 97 T + p^{2} T^{2} )^{2}( 1 + 97 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 13918 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−10.50981825181986785872294517575, −10.39047092837333457221418145916, −10.15596425832831174120485415096, −9.840655984491762970981446057829, −9.117818490591680144961085547601, −9.050575231139934831003180986404, −8.731207446676533640624337193978, −8.414257734253821439877396532851, −8.028262890006306496922678387450, −7.83918255447649374644294706000, −7.39831720023179465024042714305, −6.85375806299344915528016160779, −6.68691100168710188210643191112, −6.55437423545580095083128906891, −6.14761488436847336722925105830, −5.65703799695834316788849309154, −5.60316886228379095988567536677, −4.51991831460946341657674476768, −4.42812840412616322515225238100, −4.38205075448899025952440477021, −3.24897179604097912545719314892, −2.88055083721362980639495000710, −2.34510386953098010494079647236, −2.22481081588170554204089271884, −0.908229494399085101493476265249,
0.908229494399085101493476265249, 2.22481081588170554204089271884, 2.34510386953098010494079647236, 2.88055083721362980639495000710, 3.24897179604097912545719314892, 4.38205075448899025952440477021, 4.42812840412616322515225238100, 4.51991831460946341657674476768, 5.60316886228379095988567536677, 5.65703799695834316788849309154, 6.14761488436847336722925105830, 6.55437423545580095083128906891, 6.68691100168710188210643191112, 6.85375806299344915528016160779, 7.39831720023179465024042714305, 7.83918255447649374644294706000, 8.028262890006306496922678387450, 8.414257734253821439877396532851, 8.731207446676533640624337193978, 9.050575231139934831003180986404, 9.117818490591680144961085547601, 9.840655984491762970981446057829, 10.15596425832831174120485415096, 10.39047092837333457221418145916, 10.50981825181986785872294517575