| L(s) = 1 | − 16·2-s + 128·4-s + 556·5-s − 8.89e3·10-s + 2.20e4·13-s − 1.63e4·16-s + 3.42e4·17-s + 7.11e4·20-s + 2.31e5·25-s − 3.52e5·26-s + 2.62e5·32-s − 5.48e5·34-s + 3.15e5·37-s − 1.76e6·41-s − 3.69e6·50-s + 2.81e6·52-s − 2.81e6·53-s + 1.06e6·61-s − 2.09e6·64-s + 1.22e7·65-s + 4.38e6·68-s + 9.28e6·73-s − 5.05e6·74-s − 9.10e6·80-s − 4.78e6·81-s + 2.82e7·82-s + 1.90e7·85-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.98·5-s − 2.81·10-s + 2.77·13-s − 16-s + 1.69·17-s + 1.98·20-s + 2.95·25-s − 3.92·26-s + 1.41·32-s − 2.39·34-s + 1.02·37-s − 3.99·41-s − 4.18·50-s + 2.77·52-s − 2.59·53-s + 0.600·61-s − 64-s + 5.52·65-s + 1.69·68-s + 2.79·73-s − 1.44·74-s − 1.98·80-s − 81-s + 5.65·82-s + 3.36·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.806984099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.806984099\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T + p^{7} T^{2} \) |
| 5 | $C_2$ | \( 1 - 556 T + p^{7} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 13108 T + p^{7} T^{2} )( 1 - 8898 T + p^{7} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 40094 T + p^{7} T^{2} )( 1 + 5816 T + p^{7} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 233230 T + p^{7} T^{2} )( 1 + 233230 T + p^{7} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 563974 T + p^{7} T^{2} )( 1 + 248316 T + p^{7} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 882568 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 798602 T + p^{7} T^{2} )( 1 + 2015212 T + p^{7} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 532572 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 5370608 T + p^{7} T^{2} )( 1 - 3917418 T + p^{7} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9246170 T + p^{7} T^{2} )( 1 + 9246170 T + p^{7} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 3818296 T + p^{7} T^{2} )( 1 + 17567406 T + p^{7} T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.98878904658663768884702821905, −16.75520893545454615585206339526, −16.14817392322647560194652974510, −15.32092970610744729844989238605, −14.28783265519472665884493908568, −13.52215987512946742893001613428, −13.51990427076908147332252019516, −12.42070219283652372986892641195, −11.14628512162971523645375543368, −10.76334303112620915233552916864, −9.797252429544666142123507030542, −9.684068590209540765400341746927, −8.532225630042284690305041447421, −8.275227131921386394716489390084, −6.74964228890560416569568618706, −6.15267675561746440316927099331, −5.24410925729165942778758030810, −3.30907914136089690084584111462, −1.64299363056142997136252761803, −1.16754566942834989161443641667,
1.16754566942834989161443641667, 1.64299363056142997136252761803, 3.30907914136089690084584111462, 5.24410925729165942778758030810, 6.15267675561746440316927099331, 6.74964228890560416569568618706, 8.275227131921386394716489390084, 8.532225630042284690305041447421, 9.684068590209540765400341746927, 9.797252429544666142123507030542, 10.76334303112620915233552916864, 11.14628512162971523645375543368, 12.42070219283652372986892641195, 13.51990427076908147332252019516, 13.52215987512946742893001613428, 14.28783265519472665884493908568, 15.32092970610744729844989238605, 16.14817392322647560194652974510, 16.75520893545454615585206339526, 16.98878904658663768884702821905
