| L(s) = 1 | − 12·5-s + 44·13-s − 64·17-s + 72·25-s − 44·29-s − 180·37-s − 132·49-s + 12·53-s + 108·61-s − 528·65-s + 158·81-s + 768·85-s − 320·97-s + 204·101-s − 108·109-s + 264·113-s − 516·125-s + 127-s + 131-s + 137-s + 139-s + 528·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.39·5-s + 3.38·13-s − 3.76·17-s + 2.87·25-s − 1.51·29-s − 4.86·37-s − 2.69·49-s + 0.226·53-s + 1.77·61-s − 8.12·65-s + 1.95·81-s + 9.03·85-s − 3.29·97-s + 2.01·101-s − 0.990·109-s + 2.33·113-s − 4.12·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.64·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04011646221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04011646221\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 158 T^{4} + p^{8} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 66 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 4318 T^{4} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 + 237794 T^{4} + p^{8} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 510 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 22 T + 242 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 770 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 90 T + 4050 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 734 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 4039202 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 4290 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 18768478 T^{4} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 54 T + 1458 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 40015202 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 9794 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 10654 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 5950 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 49731358 T^{4} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 14942 T^{2} + p^{4} T^{4} )^{2} \) |
| 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
−7.57597258830492383807327133540, −7.54853613698685511045702669310, −7.07725392349658153318010529155, −6.72264032308856853556552498568, −6.68552190079505585890710017641, −6.65658579276175049794911750493, −6.49747515146291143339470101339, −5.83404560762190971618297844021, −5.78086779509900992389016107182, −5.30195765597894669867086017472, −5.17974964032379071218917657533, −4.70662985065817677716647759202, −4.46675002445122875364355192226, −4.31454609943546866067006335991, −3.94040246981922963771517878001, −3.64541068530052641328114532314, −3.51526061766362024626600168371, −3.47143561555121426697351096130, −3.09134284291728256132610567207, −2.34252644042663735235786688939, −2.06430546848504947137366532250, −1.64878202533396309353715516018, −1.40285031458551681848984117005, −0.58993053524057942994212409040, −0.05631299581672716715515250943,
0.05631299581672716715515250943, 0.58993053524057942994212409040, 1.40285031458551681848984117005, 1.64878202533396309353715516018, 2.06430546848504947137366532250, 2.34252644042663735235786688939, 3.09134284291728256132610567207, 3.47143561555121426697351096130, 3.51526061766362024626600168371, 3.64541068530052641328114532314, 3.94040246981922963771517878001, 4.31454609943546866067006335991, 4.46675002445122875364355192226, 4.70662985065817677716647759202, 5.17974964032379071218917657533, 5.30195765597894669867086017472, 5.78086779509900992389016107182, 5.83404560762190971618297844021, 6.49747515146291143339470101339, 6.65658579276175049794911750493, 6.68552190079505585890710017641, 6.72264032308856853556552498568, 7.07725392349658153318010529155, 7.54853613698685511045702669310, 7.57597258830492383807327133540
