| L(s) = 1 | + 128·4-s + 1.16e3·9-s + 3.49e3·11-s + 4.09e3·16-s − 4.45e4·29-s + 1.48e5·36-s + 4.47e5·44-s + 1.64e5·49-s − 5.24e5·64-s − 7.37e5·71-s + 2.13e6·79-s − 4.67e4·81-s + 4.06e6·99-s − 7.96e5·109-s − 5.70e6·116-s + 5.52e5·121-s + 127-s + 131-s + 137-s + 139-s + 4.76e6·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9.51e6·169-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.59·9-s + 2.62·11-s + 16-s − 1.82·29-s + 3.19·36-s + 5.25·44-s + 1.39·49-s − 2·64-s − 2.06·71-s + 4.33·79-s − 0.0878·81-s + 4.19·99-s − 0.615·109-s − 3.65·116-s + 0.311·121-s + 1.59·144-s − 1.97·169-s + 2.62·176-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(6.653921369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.653921369\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 3358 p^{2} T^{2} + p^{12} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - p^{6} T^{2} + p^{12} T^{4} )^{2} \) |
| 3 | $C_2^2$ | \( ( 1 - 194 p T^{2} + p^{12} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 874 T + p^{6} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 4755578 T^{2} + p^{12} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 748834 p T^{2} + p^{12} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 84379322 T^{2} + p^{12} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 517246 p^{2} T^{2} + p^{12} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 11146 T + p^{6} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 1020892802 T^{2} + p^{12} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 5122440814 T^{2} + p^{12} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 6189133442 T^{2} + p^{12} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 11655635374 T^{2} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 16309886018 T^{2} + p^{12} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 38490845422 T^{2} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 71539567322 T^{2} + p^{12} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 27356175482 T^{2} + p^{12} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 64347874226 T^{2} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 184406 T + p^{6} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 298950552578 T^{2} + p^{12} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 534934 T + p^{6} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 142873131578 T^{2} + p^{12} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 597656180162 T^{2} + p^{12} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 1002631840898 T^{2} + p^{12} T^{4} )^{2} \) |
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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
−7.977080964063769988648326578073, −7.72299444733545873911072127267, −7.38483739201434069903887512650, −7.27217804744979853404145449753, −7.00234687007896923075283110416, −6.92215358678522824742977019177, −6.36053569312475473042596509301, −6.32174385555022471141391065892, −6.21649879758349272648325429311, −5.80879297904344334144896942372, −5.17826750132671689963991268422, −5.13944659226290204984080756713, −4.51555146591242924341195435438, −4.16769319263295211113420047905, −3.92028164386999703615129336932, −3.88836264363144132081001104587, −3.26559331262990625610898073738, −3.04146759266038904172134060548, −2.32957620048323014948641659550, −2.21244727971885009120963743946, −1.85384745739614393558946802361, −1.50897358098949118814520307482, −1.10413056847296055875151380573, −1.09208554416119524417025916910, −0.23405174987772372526809353599,
0.23405174987772372526809353599, 1.09208554416119524417025916910, 1.10413056847296055875151380573, 1.50897358098949118814520307482, 1.85384745739614393558946802361, 2.21244727971885009120963743946, 2.32957620048323014948641659550, 3.04146759266038904172134060548, 3.26559331262990625610898073738, 3.88836264363144132081001104587, 3.92028164386999703615129336932, 4.16769319263295211113420047905, 4.51555146591242924341195435438, 5.13944659226290204984080756713, 5.17826750132671689963991268422, 5.80879297904344334144896942372, 6.21649879758349272648325429311, 6.32174385555022471141391065892, 6.36053569312475473042596509301, 6.92215358678522824742977019177, 7.00234687007896923075283110416, 7.27217804744979853404145449753, 7.38483739201434069903887512650, 7.72299444733545873911072127267, 7.977080964063769988648326578073
