| L(s) = 1 | + 12·13-s + 36·17-s + 56·29-s − 60·37-s − 4·41-s + 86·49-s + 204·53-s − 148·61-s − 264·73-s − 28·89-s − 48·97-s + 40·101-s + 172·109-s + 252·113-s + 230·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 230·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 0.923·13-s + 2.11·17-s + 1.93·29-s − 1.62·37-s − 0.0975·41-s + 1.75·49-s + 3.84·53-s − 2.42·61-s − 3.61·73-s − 0.314·89-s − 0.494·97-s + 0.396·101-s + 1.57·109-s + 2.23·113-s + 1.90·121-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 − 1.36·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.691937310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.691937310\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 86 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 230 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 530 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1010 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 28 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 430 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 1346 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 102 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3130 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 430 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 10034 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 132 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1682 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 94 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
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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
−8.511492291424585822294142141597, −8.471586395019903733861611061277, −7.76744785073590887912818600364, −7.35800887126743810241726970634, −7.13510498956268069826642411920, −6.93205715684423652147517878278, −6.03410125758577726281222296702, −5.98967888324757827104002869350, −5.70852349924646482483168129943, −5.25529430232294234075983690905, −4.68890965194278700370934288268, −4.41180512689336631050965741628, −3.83116764034511162739600995496, −3.51373730136884665569110507632, −2.97720012926039532291758795917, −2.79460632086365903987952345631, −1.97765234702309079288935450320, −1.48233592568653791111366710688, −0.937262085053607041183442641493, −0.57328307799798447391215562546,
0.57328307799798447391215562546, 0.937262085053607041183442641493, 1.48233592568653791111366710688, 1.97765234702309079288935450320, 2.79460632086365903987952345631, 2.97720012926039532291758795917, 3.51373730136884665569110507632, 3.83116764034511162739600995496, 4.41180512689336631050965741628, 4.68890965194278700370934288268, 5.25529430232294234075983690905, 5.70852349924646482483168129943, 5.98967888324757827104002869350, 6.03410125758577726281222296702, 6.93205715684423652147517878278, 7.13510498956268069826642411920, 7.35800887126743810241726970634, 7.76744785073590887912818600364, 8.471586395019903733861611061277, 8.511492291424585822294142141597
