| L(s) = 1 | + 2·3-s − 6·7-s + 5·9-s + 2·11-s − 4·13-s + 6·17-s − 6·19-s − 12·21-s + 6·23-s − 4·25-s + 10·27-s − 6·29-s + 4·33-s − 2·37-s − 8·39-s + 6·41-s + 6·43-s + 24·47-s + 21·49-s + 12·51-s − 12·57-s − 6·59-s − 14·61-s − 30·63-s − 18·67-s + 12·69-s + 6·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2.26·7-s + 5/3·9-s + 0.603·11-s − 1.10·13-s + 1.45·17-s − 1.37·19-s − 2.61·21-s + 1.25·23-s − 4/5·25-s + 1.92·27-s − 1.11·29-s + 0.696·33-s − 0.328·37-s − 1.28·39-s + 0.937·41-s + 0.914·43-s + 3.50·47-s + 3·49-s + 1.68·51-s − 1.58·57-s − 0.781·59-s − 1.79·61-s − 3.77·63-s − 2.19·67-s + 1.44·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.299402127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.299402127\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 15 T^{2} + 6 p T^{3} + 20 p T^{4} + 6 p^{2} T^{5} + 15 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - T^{2} + 34 T^{3} - 140 T^{4} + 34 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 6 p T^{5} + p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 7 T^{2} - 54 T^{3} - 204 T^{4} - 54 p T^{5} + 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T - T^{2} + 54 T^{3} + 12 T^{4} + 54 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T + T^{2} - 138 T^{3} - 660 T^{4} - 138 p T^{5} + p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2 T + T^{2} - 142 T^{3} - 1508 T^{4} - 142 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - 47 T^{2} - 6 T^{3} + 3732 T^{4} - 6 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T - 9 T^{2} + 246 T^{3} - 1372 T^{4} + 246 p T^{5} - 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T - 73 T^{2} - 54 T^{3} + 6276 T^{4} - 54 p T^{5} - 73 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 18 T + 127 T^{2} + 1134 T^{3} + 12612 T^{4} + 1134 p T^{5} + 127 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T - 97 T^{2} + 54 T^{3} + 10092 T^{4} + 54 p T^{5} - 97 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 - 18 T + 97 T^{2} - 882 T^{3} + 14772 T^{4} - 882 p T^{5} + 97 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.122432780260744564128465704578, −7.903609832611024544152735507094, −7.36033793133985790473398221516, −7.35168825284763161452944805537, −7.33462485768820063904887853516, −7.11120684372679080316926938719, −6.46960408175150434653360802389, −6.39630901918497497806049190205, −6.38828120616388474176376717509, −5.81696401799114879105412283180, −5.67336017977472620843637260124, −5.45303269820738617530664008474, −4.92692760236152258859330306472, −4.67384186769032496677940069073, −4.26256629918312557751062620821, −4.04840041830719252535537556835, −3.71599974571601652190490860191, −3.70892945981940961946842011871, −3.07031960212961596442441316760, −2.96603595367112199857029046969, −2.47081857615985642349475466340, −2.44918310669262376538627162911, −1.76323639653083310184307739574, −1.25426686852587588783194986356, −0.54647633518433370662244596702,
0.54647633518433370662244596702, 1.25426686852587588783194986356, 1.76323639653083310184307739574, 2.44918310669262376538627162911, 2.47081857615985642349475466340, 2.96603595367112199857029046969, 3.07031960212961596442441316760, 3.70892945981940961946842011871, 3.71599974571601652190490860191, 4.04840041830719252535537556835, 4.26256629918312557751062620821, 4.67384186769032496677940069073, 4.92692760236152258859330306472, 5.45303269820738617530664008474, 5.67336017977472620843637260124, 5.81696401799114879105412283180, 6.38828120616388474176376717509, 6.39630901918497497806049190205, 6.46960408175150434653360802389, 7.11120684372679080316926938719, 7.33462485768820063904887853516, 7.35168825284763161452944805537, 7.36033793133985790473398221516, 7.903609832611024544152735507094, 8.122432780260744564128465704578
