| L(s) = 1 | − 8·4-s + 56·7-s + 54·9-s + 48·16-s + 226·25-s − 448·28-s − 432·36-s + 364·37-s − 1.92e3·43-s + 1.66e3·49-s + 3.02e3·63-s − 256·64-s − 676·67-s + 680·79-s + 2.18e3·81-s − 1.80e3·100-s − 1.59e3·109-s + 2.68e3·112-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 2.59e3·144-s − 2.91e3·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s + 3.02·7-s + 2·9-s + 3/4·16-s + 1.80·25-s − 3.02·28-s − 2·36-s + 1.61·37-s − 6.83·43-s + 34/7·49-s + 6.04·63-s − 1/2·64-s − 1.23·67-s + 0.968·79-s + 3·81-s − 1.80·100-s − 1.39·109-s + 2.26·112-s − 0.181·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 3/2·144-s − 1.61·148-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.148462651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.148462651\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 113 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 1127 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 6754 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 13643 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 24298 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 33649 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 p^{2} T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 91 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 127042 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 482 T + p^{3} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 81571 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 153830 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 136229 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 384650 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 169 T + p^{3} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 625822 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 560951 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 170 T + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 281686 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 920290 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 795158 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−7.62556798949889604005979389426, −7.52343371354391101311323522999, −6.97426779331711085108138079901, −6.84025513504948653450767034107, −6.76223347052927936169615400510, −6.27545419933532043828214776169, −6.24155511387843825791081150367, −5.55202437934305944014706164388, −5.30489612428830227208101433980, −5.07372323187219120390755906897, −4.99175988225554099306624764162, −4.76852838576182832355938641514, −4.51975754585550699253645469569, −4.36251467496417748264986043154, −4.07733878585082143989048434650, −3.61716697394725447870428359699, −3.35316776841408578104892584654, −3.13048272567915659123629706527, −2.38770503717668590054075697766, −2.22494253765493218890943818821, −1.53843165010242645140709977205, −1.46531068190974400467272026790, −1.42491747555472185181717462355, −0.903275990765109192258013941994, −0.31098241857833187705129979984,
0.31098241857833187705129979984, 0.903275990765109192258013941994, 1.42491747555472185181717462355, 1.46531068190974400467272026790, 1.53843165010242645140709977205, 2.22494253765493218890943818821, 2.38770503717668590054075697766, 3.13048272567915659123629706527, 3.35316776841408578104892584654, 3.61716697394725447870428359699, 4.07733878585082143989048434650, 4.36251467496417748264986043154, 4.51975754585550699253645469569, 4.76852838576182832355938641514, 4.99175988225554099306624764162, 5.07372323187219120390755906897, 5.30489612428830227208101433980, 5.55202437934305944014706164388, 6.24155511387843825791081150367, 6.27545419933532043828214776169, 6.76223347052927936169615400510, 6.84025513504948653450767034107, 6.97426779331711085108138079901, 7.52343371354391101311323522999, 7.62556798949889604005979389426
