| L(s) = 1 | − 2·2-s − 3·3-s + 4-s − 6·5-s + 6·6-s − 12·7-s + 2·8-s + 2·9-s + 12·10-s − 3·12-s − 5·13-s + 24·14-s + 18·15-s − 4·16-s − 3·17-s − 4·18-s − 6·20-s + 36·21-s − 12·23-s − 6·24-s + 17·25-s + 10·26-s + 3·27-s − 12·28-s − 36·30-s + 2·32-s + 6·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 1/2·4-s − 2.68·5-s + 2.44·6-s − 4.53·7-s + 0.707·8-s + 2/3·9-s + 3.79·10-s − 0.866·12-s − 1.38·13-s + 6.41·14-s + 4.64·15-s − 16-s − 0.727·17-s − 0.942·18-s − 1.34·20-s + 7.85·21-s − 2.50·23-s − 1.22·24-s + 17/5·25-s + 1.96·26-s + 0.577·27-s − 2.26·28-s − 6.57·30-s + 0.353·32-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 + 5 T + T^{2} - 10 T^{3} + 82 T^{4} - 10 p T^{5} + p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 3 T - 19 T^{2} - 18 T^{3} + 342 T^{4} - 18 p T^{5} - 19 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 1620 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - T^{2} - 1716 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 - 49 T^{2} + 720 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 10686 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 9 T + 137 T^{2} - 990 T^{3} + 10722 T^{4} - 990 p T^{5} + 137 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 30 T + 482 T^{2} + 5460 T^{3} + 47343 T^{4} + 5460 p T^{5} + 482 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 9 T + 131 T^{2} + 936 T^{3} + 8742 T^{4} + 936 p T^{5} + 131 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 50 T^{2} - 228 T^{3} - 2241 T^{4} - 228 p T^{5} + 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T - 82 T^{2} + 144 T^{3} + 6327 T^{4} + 144 p T^{5} - 82 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T + 74 T^{2} - 372 T^{3} - 1449 T^{4} - 372 p T^{5} + 74 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 24 T + 362 T^{2} + 4080 T^{3} + 37947 T^{4} + 4080 p T^{5} + 362 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 39 T + 809 T^{2} + 11778 T^{3} + 128406 T^{4} + 11778 p T^{5} + 809 p^{2} T^{6} + 39 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 7 T + 132 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.894512461875042411245339024924, −8.495237136544163940562244010178, −8.201838672917920221575809326230, −8.124948522730085121833255073320, −7.68156597875810815462165669009, −7.54642146519005609743315884400, −7.14638913893938203619892257692, −7.08054401987464178557179478376, −6.89633244033457933420439212749, −6.43710549241103880379807921398, −6.39664827871028382805597708233, −6.11740602243252145076170784437, −5.82907691661979055338889169674, −5.60981092847855310671001524995, −5.34272215297827295424750096398, −4.59102875602102777040626949036, −4.46377013943940548994635233799, −4.35166462100043483663066563548, −3.90912450530183781296641046838, −3.55242703230157424368167477410, −3.36293840380481594562593739434, −3.12616451694725125542948197612, −2.55310379476121640911379277657, −2.40446390556957536163841765474, −1.19940740588260723906817380778, 0, 0, 0, 0,
1.19940740588260723906817380778, 2.40446390556957536163841765474, 2.55310379476121640911379277657, 3.12616451694725125542948197612, 3.36293840380481594562593739434, 3.55242703230157424368167477410, 3.90912450530183781296641046838, 4.35166462100043483663066563548, 4.46377013943940548994635233799, 4.59102875602102777040626949036, 5.34272215297827295424750096398, 5.60981092847855310671001524995, 5.82907691661979055338889169674, 6.11740602243252145076170784437, 6.39664827871028382805597708233, 6.43710549241103880379807921398, 6.89633244033457933420439212749, 7.08054401987464178557179478376, 7.14638913893938203619892257692, 7.54642146519005609743315884400, 7.68156597875810815462165669009, 8.124948522730085121833255073320, 8.201838672917920221575809326230, 8.495237136544163940562244010178, 8.894512461875042411245339024924
