| L(s) = 1 | + 8·4-s + 146·7-s − 190·13-s − 1.25e3·19-s − 98·25-s + 1.16e3·28-s + 1.91e3·31-s − 1.54e3·37-s − 5.09e3·43-s + 1.01e4·49-s − 1.52e3·52-s − 1.12e4·61-s − 512·64-s − 46·67-s + 2.61e4·73-s − 1.00e4·76-s + 1.22e4·79-s − 2.77e4·91-s − 1.98e4·97-s − 784·100-s − 1.24e4·103-s − 6.39e4·109-s − 482·121-s + 1.53e4·124-s + 127-s + 131-s − 1.82e5·133-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 2.97·7-s − 1.12·13-s − 3.46·19-s − 0.156·25-s + 1.48·28-s + 1.99·31-s − 1.12·37-s − 2.75·43-s + 4.21·49-s − 0.562·52-s − 3.01·61-s − 1/8·64-s − 0.0102·67-s + 4.89·73-s − 1.73·76-s + 1.96·79-s − 3.34·91-s − 2.11·97-s − 0.0783·100-s − 1.17·103-s − 5.38·109-s − 0.0329·121-s + 0.996·124-s + 6.20e−5·127-s + 5.82e−5·131-s − 10.3·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4408261343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4408261343\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 98 T^{2} - 381021 T^{4} + 98 p^{8} T^{6} + p^{16} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 73 T + 2928 T^{2} - 73 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 482 T^{2} - 214126557 T^{4} + 482 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 95 T - 19536 T^{2} + 95 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 156674 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 313 T + p^{4} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 143810 T^{2} - 57629669181 T^{4} + 143810 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 1023070 T^{2} + 546425811939 T^{4} - 1023070 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 958 T - 5757 T^{2} - 958 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 385 T + p^{4} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 6722 T^{2} - 7984880043837 T^{4} + 6722 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 2546 T + 3063315 T^{2} + 2546 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 9730562 T^{2} + 70872550174083 T^{4} + 9730562 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 8772194 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 17045090 T^{2} + 143704655503779 T^{4} + 17045090 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 5615 T + 17682384 T^{2} + 5615 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 23 T - 20150592 T^{2} + 23 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 50657474 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6527 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 6121 T - 1483440 T^{2} - 6121 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 88608290 T^{2} + 5599136824585059 T^{4} + 88608290 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 117926210 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 9935 T + 10174944 T^{2} + 9935 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−8.655105038891684647015287678345, −8.200451196531010095618231569625, −8.184960686484954336493170214693, −7.929509584614280690872137547404, −7.83562242916161376069845054129, −7.54575223180545269499180414357, −6.65434939673106193954208561416, −6.65091265421645404703239766548, −6.60630969447404824823535901998, −6.53317109667125518286399229025, −5.64270843611752939955206296856, −5.29345057046917521136651281567, −5.25309457204953267548968520326, −4.74344767172991247721066279009, −4.63519483883296092530856630441, −4.35607310065816430999389249054, −3.96631482477220987485079021314, −3.60177966569885866304210290719, −2.77543587310433146049099774403, −2.60483379679315000410542984555, −2.04066066296330912114092422081, −1.86414697714390758711621407761, −1.57976773438997981869896282800, −1.01381142596655451604013602844, −0.10129198398270656946390178237,
0.10129198398270656946390178237, 1.01381142596655451604013602844, 1.57976773438997981869896282800, 1.86414697714390758711621407761, 2.04066066296330912114092422081, 2.60483379679315000410542984555, 2.77543587310433146049099774403, 3.60177966569885866304210290719, 3.96631482477220987485079021314, 4.35607310065816430999389249054, 4.63519483883296092530856630441, 4.74344767172991247721066279009, 5.25309457204953267548968520326, 5.29345057046917521136651281567, 5.64270843611752939955206296856, 6.53317109667125518286399229025, 6.60630969447404824823535901998, 6.65091265421645404703239766548, 6.65434939673106193954208561416, 7.54575223180545269499180414357, 7.83562242916161376069845054129, 7.929509584614280690872137547404, 8.184960686484954336493170214693, 8.200451196531010095618231569625, 8.655105038891684647015287678345
