| L(s) = 1 | + 14·7-s + 92·13-s − 2·19-s − 32·25-s − 98·31-s − 34·37-s − 188·43-s + 49·49-s + 80·61-s + 46·67-s − 34·73-s − 158·79-s + 1.28e3·91-s − 160·97-s + 46·103-s − 142·109-s + 208·121-s + 127-s + 131-s − 28·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·7-s + 7.07·13-s − 0.105·19-s − 1.27·25-s − 3.16·31-s − 0.918·37-s − 4.37·43-s + 49-s + 1.31·61-s + 0.686·67-s − 0.465·73-s − 2·79-s + 14.1·91-s − 1.64·97-s + 0.446·103-s − 1.30·109-s + 1.71·121-s + 0.00787·127-s + 0.00763·131-s − 0.210·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.098396311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.098396311\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 32 T^{2} + 399 T^{4} + 32 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 208 T^{2} + 28623 T^{4} - 208 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 290 T^{2} + 579 T^{4} + 290 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + T - 360 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 770 T^{2} + 313059 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 49 T + 1440 T^{2} + 49 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 17 T - 1080 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2912 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 47 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 2960 T^{2} + 3881919 T^{4} + 2960 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 1582 T^{2} - 5387757 T^{4} - 1582 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 4370 T^{2} + 6979539 T^{4} + 4370 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 40 T - 2121 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 23 T - 3960 T^{2} - 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 6032 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 17 T - 5040 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{4}( 1 - p T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 2528 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 2590 T^{2} - 56034141 T^{4} - 2590 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
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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
−6.85886707040224696681206280344, −6.71112332183913033110448324858, −6.39887313091552205671213153166, −6.29213762819488196518202065622, −5.91782199641265645199256863764, −5.66569265907299854544553028429, −5.64926225531785140505073396526, −5.60585303720514930301112748131, −5.12032359666615993612800738529, −4.91909357842074048255224775997, −4.57904563935108502431845680779, −4.18232806504292397265300267415, −4.02585206918778337785281556502, −3.68455445482901524178133188513, −3.67139399976482296891075856919, −3.54094889400266365484414886294, −3.16365209515022091305171681963, −3.03942414869655281741441763602, −2.04549936382176454066557562292, −1.95407775186069141000352699009, −1.60700679005466256741153799480, −1.45550120393563986383252144314, −1.37839573157699401027098016248, −0.982379228572056584076253720844, −0.29709721381251255088277996491,
0.29709721381251255088277996491, 0.982379228572056584076253720844, 1.37839573157699401027098016248, 1.45550120393563986383252144314, 1.60700679005466256741153799480, 1.95407775186069141000352699009, 2.04549936382176454066557562292, 3.03942414869655281741441763602, 3.16365209515022091305171681963, 3.54094889400266365484414886294, 3.67139399976482296891075856919, 3.68455445482901524178133188513, 4.02585206918778337785281556502, 4.18232806504292397265300267415, 4.57904563935108502431845680779, 4.91909357842074048255224775997, 5.12032359666615993612800738529, 5.60585303720514930301112748131, 5.64926225531785140505073396526, 5.66569265907299854544553028429, 5.91782199641265645199256863764, 6.29213762819488196518202065622, 6.39887313091552205671213153166, 6.71112332183913033110448324858, 6.85886707040224696681206280344
