| L(s) = 1 | + 32·31-s + 8·61-s − 9·81-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | + 5.74·31-s + 1.02·61-s − 81-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.651257024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.651257024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 382 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 98 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 4222 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 1778 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 9938 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
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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
−6.86283374453819901317107383083, −6.66709908510399106861920580891, −6.62786762363415075786743681736, −6.31986026313351043401535364921, −6.21576872336651521318688874211, −5.77290692144368397795452045057, −5.71575721077134497730604722454, −5.45014106752503096341868284215, −5.19727444345937182614754279955, −4.76081794227173862564391240751, −4.62264147842176870284886006498, −4.59657720775064796326895936823, −4.25429214675913269341857021785, −3.95500394146663808701560697744, −3.91286719601032352991475810704, −3.24587447288801895800646260029, −2.97907138843973790078088287555, −2.90068815041112368901336445051, −2.88720708027315829855884471091, −2.25771757618272196111389944321, −2.00221240568426691421000120027, −1.73580245317058583049051031545, −1.14226870240276052707038592325, −0.71255600069348229125914105221, −0.67146782116457868039666048247,
0.67146782116457868039666048247, 0.71255600069348229125914105221, 1.14226870240276052707038592325, 1.73580245317058583049051031545, 2.00221240568426691421000120027, 2.25771757618272196111389944321, 2.88720708027315829855884471091, 2.90068815041112368901336445051, 2.97907138843973790078088287555, 3.24587447288801895800646260029, 3.91286719601032352991475810704, 3.95500394146663808701560697744, 4.25429214675913269341857021785, 4.59657720775064796326895936823, 4.62264147842176870284886006498, 4.76081794227173862564391240751, 5.19727444345937182614754279955, 5.45014106752503096341868284215, 5.71575721077134497730604722454, 5.77290692144368397795452045057, 6.21576872336651521318688874211, 6.31986026313351043401535364921, 6.62786762363415075786743681736, 6.66709908510399106861920580891, 6.86283374453819901317107383083
