| L(s) = 1 | − 4·2-s + 8·3-s + 8·4-s − 32·6-s − 12·7-s − 8·8-s + 16·9-s + 4·11-s + 64·12-s + 48·14-s − 4·16-s − 64·18-s + 24·19-s − 96·21-s − 16·22-s − 64·24-s − 56·27-s − 96·28-s + 76·29-s + 60·31-s + 32·32-s + 32·33-s + 128·36-s + 68·37-s − 96·38-s + 64·41-s + 384·42-s + ⋯ |
| L(s) = 1 | − 2·2-s + 8/3·3-s + 2·4-s − 5.33·6-s − 1.71·7-s − 8-s + 16/9·9-s + 4/11·11-s + 16/3·12-s + 24/7·14-s − 1/4·16-s − 3.55·18-s + 1.26·19-s − 4.57·21-s − 0.727·22-s − 8/3·24-s − 2.07·27-s − 3.42·28-s + 2.62·29-s + 1.93·31-s + 32-s + 0.969·33-s + 32/9·36-s + 1.83·37-s − 2.52·38-s + 1.56·41-s + 64/7·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.369660550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.369660550\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $D_{4}$ | \( ( 1 - 4 T + 16 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 480 T^{3} + 3119 T^{4} + 480 p^{2} T^{5} + 72 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 480 T^{3} - 29281 T^{4} + 480 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 706 T^{2} + 283875 T^{4} - 706 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 656 T^{2} + 521250 T^{4} - 656 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 38 T + 1179 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 60 T + 1800 T^{2} - 83040 T^{3} + 3651983 T^{4} - 83040 p^{2} T^{5} + 1800 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 68 T + 2312 T^{2} - 103020 T^{3} + 4569134 T^{4} - 103020 p^{2} T^{5} + 2312 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 64 T + 2048 T^{2} - 29760 T^{3} - 1046206 T^{4} - 29760 p^{2} T^{5} + 2048 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 3536 T^{2} + 7156290 T^{4} - 3536 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} + 66912 T^{3} + 1577327 T^{4} + 66912 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 130 T + 9627 T^{2} - 130 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 52 T + 1352 T^{2} - 28704 T^{3} - 7969633 T^{4} - 28704 p^{2} T^{5} + 1352 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 50 T + 6123 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 44 T + 968 T^{2} - 7392 T^{3} - 18614593 T^{4} - 7392 p^{2} T^{5} + 968 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 56 T + 1568 T^{2} + 82824 T^{3} - 38135518 T^{4} + 82824 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 36 T + 648 T^{2} + 51156 T^{3} - 41524018 T^{4} + 51156 p^{2} T^{5} + 648 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 32 T + 9282 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 140 T + 9800 T^{2} - 850080 T^{3} + 73070879 T^{4} - 850080 p^{2} T^{5} + 9800 p^{4} T^{6} - 140 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 84 T + 3528 T^{2} - 703164 T^{3} + 139944782 T^{4} - 703164 p^{2} T^{5} + 3528 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 128 T + 8192 T^{2} + 1245312 T^{3} + 189204482 T^{4} + 1245312 p^{2} T^{5} + 8192 p^{4} T^{6} + 128 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−7.81197095110562372757576484235, −7.27152519524731824078023587661, −6.96360495589784971016362869947, −6.89958857788135326461171161697, −6.60051061002178159112726842337, −6.38165681743021923668050946598, −6.17098340961632319326297954163, −5.72488741840994912485505582894, −5.69393507973294220170367814025, −5.28317085063561384055071703613, −4.83106263903589045031246530529, −4.68231494132880537791801483997, −3.98032138486542581456516078824, −3.89750308855399098014692962347, −3.89676859123201423275823220495, −3.41336365950101708746110089076, −2.86093280876958647625930752539, −2.74721111229795710243519719486, −2.66134003832203711050747751742, −2.54405498473811401529857515840, −2.33867477214843341466125954638, −1.48672494519905420276215315122, −1.02837584295240086607803036902, −0.806427281464001714615918895965, −0.46187142096878385648108729951,
0.46187142096878385648108729951, 0.806427281464001714615918895965, 1.02837584295240086607803036902, 1.48672494519905420276215315122, 2.33867477214843341466125954638, 2.54405498473811401529857515840, 2.66134003832203711050747751742, 2.74721111229795710243519719486, 2.86093280876958647625930752539, 3.41336365950101708746110089076, 3.89676859123201423275823220495, 3.89750308855399098014692962347, 3.98032138486542581456516078824, 4.68231494132880537791801483997, 4.83106263903589045031246530529, 5.28317085063561384055071703613, 5.69393507973294220170367814025, 5.72488741840994912485505582894, 6.17098340961632319326297954163, 6.38165681743021923668050946598, 6.60051061002178159112726842337, 6.89958857788135326461171161697, 6.96360495589784971016362869947, 7.27152519524731824078023587661, 7.81197095110562372757576484235
