| L(s) = 1 | − 2·5-s + 2·7-s + 2·11-s − 2·13-s − 16·19-s + 6·23-s + 11·25-s + 10·29-s + 10·31-s − 4·35-s + 16·37-s − 14·41-s + 10·43-s − 2·47-s + 9·49-s − 16·53-s − 4·55-s − 14·59-s + 6·61-s + 4·65-s + 10·67-s − 8·71-s + 4·77-s + 22·79-s + 6·83-s + 32·89-s − 4·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 0.603·11-s − 0.554·13-s − 3.67·19-s + 1.25·23-s + 11/5·25-s + 1.85·29-s + 1.79·31-s − 0.676·35-s + 2.63·37-s − 2.18·41-s + 1.52·43-s − 0.291·47-s + 9/7·49-s − 2.19·53-s − 0.539·55-s − 1.82·59-s + 0.768·61-s + 0.496·65-s + 1.22·67-s − 0.949·71-s + 0.455·77-s + 2.47·79-s + 0.658·83-s + 3.39·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.594862183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.594862183\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 2 T - 5 T^{2} + 10 T^{3} + 4 T^{4} + 10 p T^{5} - 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 13 T^{2} + 10 T^{3} + 124 T^{4} + 10 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 T + T^{2} - 46 T^{3} - 212 T^{4} - 46 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T - 13 T^{2} - 18 T^{3} + 1044 T^{4} - 18 p T^{5} - 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 10 T + 41 T^{2} - 10 T^{3} - 260 T^{4} - 10 p T^{5} + 41 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 10 T + 19 T^{2} - 190 T^{3} + 2500 T^{4} - 190 p T^{5} + 19 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 14 T + 89 T^{2} + 350 T^{3} + 1732 T^{4} + 350 p T^{5} + 89 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 47 | $D_4\times C_2$ | \( 1 + 2 T - 37 T^{2} - 106 T^{3} - 716 T^{4} - 106 p T^{5} - 37 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 14 T + 83 T^{2} - 70 T^{3} - 2276 T^{4} - 70 p T^{5} + 83 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 71 T^{2} + 90 T^{3} + 5532 T^{4} + 90 p T^{5} - 71 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 10 T - 5 T^{2} + 290 T^{3} - 164 T^{4} + 290 p T^{5} - 5 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 22 T + 211 T^{2} - 2530 T^{3} + 30052 T^{4} - 2530 p T^{5} + 211 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 18 p T^{5} - 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 16 T + 218 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 167 T^{2} - 46 T^{3} + 19444 T^{4} - 46 p T^{5} - 167 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.42268808209194544749927564598, −6.89312993571374996351988991092, −6.80340243471403702201606526009, −6.58179318671794532247993802529, −6.39680425641708513216299319097, −6.28440302381861851082274766102, −6.00837561683279368499912169913, −5.96528859910583408440556044370, −5.13943101309279274382693488963, −5.00641231546535804720315721015, −4.77294621327732342571617056305, −4.67335171009207551398331777931, −4.63557552129580965839884709151, −4.35939611524518985481579969079, −3.79730992822958363113529574681, −3.76767012720314649028328904312, −3.50716269744742240297203477266, −2.81691601940422826164065540518, −2.79934748383665522844416444455, −2.59829802849472811912936849620, −2.06788885321750734481405473282, −1.96527873155203264897516919106, −1.14262324060659793584923036025, −1.00254583658126544254677758887, −0.47841581401103970925867896547,
0.47841581401103970925867896547, 1.00254583658126544254677758887, 1.14262324060659793584923036025, 1.96527873155203264897516919106, 2.06788885321750734481405473282, 2.59829802849472811912936849620, 2.79934748383665522844416444455, 2.81691601940422826164065540518, 3.50716269744742240297203477266, 3.76767012720314649028328904312, 3.79730992822958363113529574681, 4.35939611524518985481579969079, 4.63557552129580965839884709151, 4.67335171009207551398331777931, 4.77294621327732342571617056305, 5.00641231546535804720315721015, 5.13943101309279274382693488963, 5.96528859910583408440556044370, 6.00837561683279368499912169913, 6.28440302381861851082274766102, 6.39680425641708513216299319097, 6.58179318671794532247993802529, 6.80340243471403702201606526009, 6.89312993571374996351988991092, 7.42268808209194544749927564598
