| L(s) = 1 | + 3.57e3·9-s + 8.33e3·17-s − 3.86e4·25-s + 2.35e5·41-s + 4.72e5·49-s − 8.87e5·73-s + 6.35e6·81-s − 1.52e6·89-s − 1.85e6·97-s + 5.09e6·113-s + 8.47e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.98e7·153-s + 157-s + 163-s + 167-s − 9.98e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4.90·9-s + 1.69·17-s − 2.47·25-s + 3.42·41-s + 4.01·49-s − 2.28·73-s + 11.9·81-s − 2.15·89-s − 2.03·97-s + 3.53·113-s + 4.78·121-s + 8.32·153-s − 2.06·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(38.20595942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(38.20595942\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( ( 1 - 596 p T^{2} + 180022 p^{2} T^{4} - 596 p^{13} T^{6} + p^{24} T^{8} )^{2} \) |
| 5 | \( ( 1 + 772 p^{2} T^{2} + 2049246 p^{3} T^{4} + 772 p^{14} T^{6} + p^{24} T^{8} )^{2} \) |
| 7 | \( ( 1 - 236356 T^{2} + 28021959366 T^{4} - 236356 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 11 | \( ( 1 - 4238012 T^{2} + 10442076641958 T^{4} - 4238012 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 13 | \( ( 1 + 4994596 T^{2} + 30260873415846 T^{4} + 4994596 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 17 | \( ( 1 - 2084 T + 32560902 T^{2} - 2084 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 19 | \( ( 1 - 184369532 T^{2} + 12923720463999078 T^{4} - 184369532 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 23 | \( ( 1 - 212117956 T^{2} + 1001159632705962 p T^{4} - 212117956 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 29 | \( ( 1 + 1409719204 T^{2} + 1160515330165289766 T^{4} + 1409719204 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2758360324 T^{2} + 3362667870952277766 T^{4} - 2758360324 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 37 | \( ( 1 + 8121202276 T^{2} + 29600495645847907686 T^{4} + 8121202276 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 41 | \( ( 1 - 58972 T + 240432438 p T^{2} - 58972 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 43 | \( ( 1 - 16632984764 T^{2} + \)\(13\!\cdots\!46\)\( T^{4} - 16632984764 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 47 | \( ( 1 - 13578767236 T^{2} + \)\(16\!\cdots\!86\)\( T^{4} - 13578767236 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 53 | \( ( 1 + 42194545636 T^{2} + \)\(11\!\cdots\!86\)\( T^{4} + 42194545636 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 59 | \( ( 1 - 131907478076 T^{2} + \)\(79\!\cdots\!26\)\( T^{4} - 131907478076 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 61 | \( ( 1 + 2637195124 p T^{2} + \)\(11\!\cdots\!46\)\( T^{4} + 2637195124 p^{13} T^{6} + p^{24} T^{8} )^{2} \) |
| 67 | \( ( 1 - 253874829308 T^{2} + \)\(30\!\cdots\!58\)\( T^{4} - 253874829308 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 71 | \( ( 1 - 164364621124 T^{2} + \)\(21\!\cdots\!06\)\( T^{4} - 164364621124 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 73 | \( ( 1 + 221956 T + 284381984742 T^{2} + 221956 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 79 | \( ( 1 - 828257339524 T^{2} + \)\(28\!\cdots\!06\)\( T^{4} - 828257339524 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 83 | \( ( 1 + 201038775172 T^{2} + \)\(21\!\cdots\!98\)\( T^{4} + 201038775172 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 89 | \( ( 1 + 380612 T + 970422538278 T^{2} + 380612 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 97 | \( ( 1 + 463388 T + 953987784774 T^{2} + 463388 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.25027081322391383865058686915, −4.19481603631090016785424372732, −3.89499015667208496323507373916, −3.89146276723002090001564483812, −3.87026180458895331417615986870, −3.58982172598759790444347788174, −3.27313808261264690460811396347, −3.20509868231752500009451107893, −3.14447242309211294834742953006, −2.68390589828608851396630409355, −2.64095567522427803494602333600, −2.45255899310010222121206104282, −2.25135598641266720527251431391, −2.02062137185212549035992421583, −1.89915768802745231017442343730, −1.66334843337599710019188149670, −1.62348878504049525785746871340, −1.47872795516758367445254104245, −1.25037136673945085541153002393, −1.07629607635726518056824546255, −0.809668858050384499440211259337, −0.792564053899169156112063696314, −0.57159501407762511949865604264, −0.54122833473784017410696098394, −0.18283820631947604492357941034,
0.18283820631947604492357941034, 0.54122833473784017410696098394, 0.57159501407762511949865604264, 0.792564053899169156112063696314, 0.809668858050384499440211259337, 1.07629607635726518056824546255, 1.25037136673945085541153002393, 1.47872795516758367445254104245, 1.62348878504049525785746871340, 1.66334843337599710019188149670, 1.89915768802745231017442343730, 2.02062137185212549035992421583, 2.25135598641266720527251431391, 2.45255899310010222121206104282, 2.64095567522427803494602333600, 2.68390589828608851396630409355, 3.14447242309211294834742953006, 3.20509868231752500009451107893, 3.27313808261264690460811396347, 3.58982172598759790444347788174, 3.87026180458895331417615986870, 3.89146276723002090001564483812, 3.89499015667208496323507373916, 4.19481603631090016785424372732, 4.25027081322391383865058686915
Plot not available for L-functions of degree greater than 10.
