| L(s) = 1 | + 4·3-s + 2·4-s + 4·9-s + 8·12-s − 5·16-s − 12·23-s − 2·25-s − 4·27-s − 24·29-s + 8·36-s − 20·43-s − 20·48-s + 20·49-s + 8·61-s − 20·64-s − 48·69-s − 8·75-s + 8·79-s − 10·81-s − 96·87-s − 24·92-s − 4·100-s − 24·101-s − 20·103-s + 36·107-s − 8·108-s + 48·113-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 4-s + 4/3·9-s + 2.30·12-s − 5/4·16-s − 2.50·23-s − 2/5·25-s − 0.769·27-s − 4.45·29-s + 4/3·36-s − 3.04·43-s − 2.88·48-s + 20/7·49-s + 1.02·61-s − 5/2·64-s − 5.77·69-s − 0.923·75-s + 0.900·79-s − 1.11·81-s − 10.2·87-s − 2.50·92-s − 2/5·100-s − 2.38·101-s − 1.97·103-s + 3.48·107-s − 0.769·108-s + 4.51·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3951359977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3951359977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | $D_{4}$ | \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 714 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 678 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 76 T^{2} + 3114 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 2166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 26874 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 17766 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.38395192315750668002601933356, −7.26747374160041593990663610613, −6.93457527284559248569112368375, −6.85873468206045005010537188332, −6.40424785885652921632956229655, −6.10268196564694529031558416917, −5.97091499516259207279901280593, −5.76250675660818695144590982684, −5.66656822945540211532878949214, −5.11433775261458911609107657041, −4.88809001428979001219866922323, −4.85998377840211082797309076469, −4.05916982647421300971515370770, −3.99353023372760538219440792833, −3.81673343901394822518910679946, −3.77417954179283042216468986387, −3.20247770031679455055116079903, −3.06309547032121625227549428416, −2.82776266815276402986537956134, −2.26142310007952345121911542415, −2.09511653932071999289730720517, −2.00758175216294599107049326855, −2.00017951303479350215762167599, −1.27545013423109265229125539164, −0.10573715502671083829674149249,
0.10573715502671083829674149249, 1.27545013423109265229125539164, 2.00017951303479350215762167599, 2.00758175216294599107049326855, 2.09511653932071999289730720517, 2.26142310007952345121911542415, 2.82776266815276402986537956134, 3.06309547032121625227549428416, 3.20247770031679455055116079903, 3.77417954179283042216468986387, 3.81673343901394822518910679946, 3.99353023372760538219440792833, 4.05916982647421300971515370770, 4.85998377840211082797309076469, 4.88809001428979001219866922323, 5.11433775261458911609107657041, 5.66656822945540211532878949214, 5.76250675660818695144590982684, 5.97091499516259207279901280593, 6.10268196564694529031558416917, 6.40424785885652921632956229655, 6.85873468206045005010537188332, 6.93457527284559248569112368375, 7.26747374160041593990663610613, 7.38395192315750668002601933356
