| L(s) = 1 | + 2-s − 3-s − 6-s − 2·7-s − 4·11-s + 4·13-s − 2·14-s − 4·17-s + 14·19-s + 2·21-s − 4·22-s + 4·23-s − 5·25-s + 4·26-s − 11·29-s + 7·31-s − 32-s + 4·33-s − 4·34-s + 20·37-s + 14·38-s − 4·39-s + 14·41-s + 2·42-s − 28·43-s + 4·46-s − 4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.408·6-s − 0.755·7-s − 1.20·11-s + 1.10·13-s − 0.534·14-s − 0.970·17-s + 3.21·19-s + 0.436·21-s − 0.852·22-s + 0.834·23-s − 25-s + 0.784·26-s − 2.04·29-s + 1.25·31-s − 0.176·32-s + 0.696·33-s − 0.685·34-s + 3.28·37-s + 2.27·38-s − 0.640·39-s + 2.18·41-s + 0.308·42-s − 4.26·43-s + 0.589·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18974736 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18974736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6900769363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6900769363\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_4$$\times$$C_4$ | \( ( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2^2:C_4$ | \( 1 + 2 T - 3 T^{2} + 10 T^{3} + 71 T^{4} + 10 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 50 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 + 4 T - T^{2} - 72 T^{3} - 271 T^{4} - 72 p T^{5} - p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 14 T + 117 T^{2} - 742 T^{3} + 3665 T^{4} - 742 p T^{5} + 117 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 11 T + 17 T^{2} - 207 T^{3} - 1420 T^{4} - 207 p T^{5} + 17 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 7 T - 7 T^{2} + 241 T^{3} - 1220 T^{4} + 241 p T^{5} - 7 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 20 T + 203 T^{2} - 1600 T^{3} + 10729 T^{4} - 1600 p T^{5} + 203 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 14 T + 95 T^{2} - 786 T^{3} + 6569 T^{4} - 786 p T^{5} + 95 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_4\times C_2$ | \( 1 + 4 T - 31 T^{2} - 312 T^{3} + 209 T^{4} - 312 p T^{5} - 31 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 5 T - 43 T^{2} - 5 p T^{3} + 1244 T^{4} - 5 p^{2} T^{5} - 43 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 530 p T^{5} + 131 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 18 T + 83 T^{2} + 66 T^{3} + 625 T^{4} + 66 p T^{5} + 83 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_4\times C_2$ | \( 1 + 6 T - 35 T^{2} - 636 T^{3} - 1331 T^{4} - 636 p T^{5} - 35 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 27 T + 251 T^{2} + 801 T^{3} + 64 T^{4} + 801 p T^{5} + 251 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 4 T - 33 T^{2} + 542 T^{3} + 8375 T^{4} + 542 p T^{5} - 33 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 2 T - 79 T^{2} - 86 T^{3} + 7139 T^{4} - 86 p T^{5} - 79 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 18 T + 254 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 6 T - 21 T^{2} + 868 T^{3} - 2331 T^{4} + 868 p T^{5} - 21 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−11.44025664610118655710534164403, −10.67210631574856062910803079323, −10.51637924901720960212812651175, −10.45192267802970947441363941606, −9.772810943573574442982284061122, −9.613273862287142170813815204638, −9.184925978065639041320695420887, −9.182679536335233007389406803725, −8.812175321490319172773735827305, −7.893444511474028259686950431622, −7.889303538009648557232702296842, −7.63301795319623868224634281725, −7.45594156774574983783933490283, −6.69713310380024019143097439822, −6.50644209949263145065873082498, −5.85928561362918358084659636449, −5.73535053188573036573677677582, −5.72883355505669511349423103519, −4.84907279068312103575574970508, −4.57781489569339494074139422872, −4.41395228321192014377078444129, −3.44652457399143454084560545978, −3.07234390707444695897808124003, −2.97038778325749746773995845452, −1.60160929682408434522196986924,
1.60160929682408434522196986924, 2.97038778325749746773995845452, 3.07234390707444695897808124003, 3.44652457399143454084560545978, 4.41395228321192014377078444129, 4.57781489569339494074139422872, 4.84907279068312103575574970508, 5.72883355505669511349423103519, 5.73535053188573036573677677582, 5.85928561362918358084659636449, 6.50644209949263145065873082498, 6.69713310380024019143097439822, 7.45594156774574983783933490283, 7.63301795319623868224634281725, 7.889303538009648557232702296842, 7.893444511474028259686950431622, 8.812175321490319172773735827305, 9.182679536335233007389406803725, 9.184925978065639041320695420887, 9.613273862287142170813815204638, 9.772810943573574442982284061122, 10.45192267802970947441363941606, 10.51637924901720960212812651175, 10.67210631574856062910803079323, 11.44025664610118655710534164403
