| L(s) = 1 | + 12·4-s − 52·9-s + 80·16-s + 896·31-s − 624·36-s − 280·41-s + 1.24e3·49-s + 192·64-s − 288·71-s + 1.85e3·79-s + 570·81-s − 1.06e3·89-s + 4.82e3·121-s + 1.07e4·124-s + 127-s + 131-s + 137-s + 139-s − 4.16e3·144-s + 149-s + 151-s + 157-s + 163-s − 3.36e3·164-s + 167-s − 3.18e3·169-s + 173-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1.92·9-s + 5/4·16-s + 5.19·31-s − 2.88·36-s − 1.06·41-s + 3.62·49-s + 3/8·64-s − 0.481·71-s + 2.64·79-s + 0.781·81-s − 1.26·89-s + 3.62·121-s + 7.78·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 2.40·144-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s − 1.59·164-s + 0.000463·167-s − 1.45·169-s + 0.000439·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.952246529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.952246529\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - 3 p^{2} T^{2} + p^{6} T^{4} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 622 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2410 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 1594 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 9630 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 12346 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 1230 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 23578 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 224 T + p^{3} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 42058 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 33878 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 94750 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 296746 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 125130 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 444890 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 571034 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 691598 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 464 T + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 846522 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 266 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 837310 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−8.701030867503481429733882334268, −8.265231492652636157862774087866, −8.021719702271845100832207587415, −7.86850817602438211414791684123, −7.80269237312966961466479748827, −7.04834304886679975608279498203, −6.90809789943188335012844106266, −6.75140857895597768604712232552, −6.54419702812773817185770952003, −5.91699982919666314271160818299, −5.88335829941382039452951668664, −5.86418345420620279456466499598, −5.34999267387999694658590908998, −4.92086115989654679130767786062, −4.47148527054231132169582418371, −4.44851949265843549853575812514, −3.81068186683113926582392619622, −3.27595981300350026494698688499, −3.09197810986066512827418445773, −2.62055683258117400334377828566, −2.54315623063400979363637155802, −2.17600841692856233911964188014, −1.48044064865723989614555377506, −0.77718893750762334175125740961, −0.60487308560625746098344417244,
0.60487308560625746098344417244, 0.77718893750762334175125740961, 1.48044064865723989614555377506, 2.17600841692856233911964188014, 2.54315623063400979363637155802, 2.62055683258117400334377828566, 3.09197810986066512827418445773, 3.27595981300350026494698688499, 3.81068186683113926582392619622, 4.44851949265843549853575812514, 4.47148527054231132169582418371, 4.92086115989654679130767786062, 5.34999267387999694658590908998, 5.86418345420620279456466499598, 5.88335829941382039452951668664, 5.91699982919666314271160818299, 6.54419702812773817185770952003, 6.75140857895597768604712232552, 6.90809789943188335012844106266, 7.04834304886679975608279498203, 7.80269237312966961466479748827, 7.86850817602438211414791684123, 8.021719702271845100832207587415, 8.265231492652636157862774087866, 8.701030867503481429733882334268
