| L(s) = 1 | − 32·2-s − 34·3-s + 256·4-s + 1.08e3·6-s + 8.19e3·8-s + 6.56e3·9-s − 2.71e4·11-s − 8.70e3·12-s − 2.62e5·16-s − 3.24e5·17-s − 2.09e5·18-s + 1.44e5·19-s + 8.69e5·22-s − 2.78e5·24-s − 7.81e5·25-s − 6.29e5·27-s + 2.09e6·32-s + 9.23e5·33-s + 1.03e7·34-s + 1.67e6·36-s − 4.62e6·38-s − 4.09e6·41-s + 5.42e6·43-s − 6.95e6·44-s + 8.91e6·48-s − 1.15e7·49-s + 2.50e7·50-s + ⋯ |
| L(s) = 1 | − 2·2-s − 0.419·3-s + 4-s + 0.839·6-s + 2·8-s + 9-s − 1.85·11-s − 0.419·12-s − 4·16-s − 3.88·17-s − 2·18-s + 1.10·19-s + 3.71·22-s − 0.839·24-s − 2·25-s − 1.18·27-s + 2·32-s + 0.778·33-s + 7.77·34-s + 36-s − 2.21·38-s − 1.45·41-s + 1.58·43-s − 1.85·44-s + 1.67·48-s − 2·49-s + 4·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.02510277291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02510277291\) |
| \(L(5)\) |
|
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 | $C_2$ | \( ( 1 + p^{4} T + p^{8} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 34 T - 5405 T^{2} + 34 p^{8} T^{3} + p^{16} T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )^{2}( 1 + p^{4} T + p^{8} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )^{2}( 1 + p^{4} T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 27166 T + p^{8} T^{2} )^{2}( 1 - 27166 T + 523632675 T^{2} - 27166 p^{8} T^{3} + p^{16} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )^{2}( 1 + p^{4} T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 162434 T + 19409046915 T^{2} + 162434 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 72286 T - 11758297245 T^{2} - 72286 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )^{2}( 1 + p^{4} T + p^{8} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )^{2}( 1 + p^{4} T + p^{8} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )^{2}( 1 + p^{4} T + p^{8} T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{4}( 1 + p^{4} T )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 4099006 T + p^{8} T^{2} )^{2}( 1 - 4099006 T + 8816924958915 T^{2} - 4099006 p^{8} T^{3} + p^{16} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5426402 T + p^{8} T^{2} )^{2}( 1 + 5426402 T + 17757638388003 T^{2} + 5426402 p^{8} T^{3} + p^{16} T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )^{2}( 1 + p^{4} T + p^{8} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{4}( 1 + p^{4} T )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + 24178078 T + p^{8} T^{2} )^{2}( 1 - 24178078 T + 437749018169763 T^{2} - 24178078 p^{8} T^{3} + p^{16} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )^{2}( 1 + p^{4} T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 13944286 T + p^{8} T^{2} )^{2}( 1 - 13944286 T - 211624565506845 T^{2} - 13944286 p^{8} T^{3} + p^{16} T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{4}( 1 + p^{4} T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 33567554 T + 320320589648835 T^{2} + 33567554 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )^{2}( 1 + p^{4} T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 30209954 T - 1339650911456925 T^{2} + 30209954 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 95519806 T + p^{8} T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 77418238 T + p^{8} T^{2} )^{2}( 1 - 77418238 T - 1843850019352317 T^{2} - 77418238 p^{8} T^{3} + p^{16} T^{4} ) \) |
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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
−8.989290538003359400676858255202, −8.923342062199955172726526362172, −8.733317417286022271261390996046, −8.113249293398086687134823507742, −7.959652229030541658060876250255, −7.65148445992319205146466931819, −7.53158066427751035955554867823, −6.97791762334062827919648444075, −6.83976838095530429117676897964, −6.54684725007864133721823545594, −5.84319643579404805015854289831, −5.57876702735004599115221210538, −5.25782778861953549805928775161, −4.63677026266559393030178963413, −4.45097609184465764700815762996, −4.26156220608959062701388963697, −4.00850385571617367165391393664, −2.94548301039826315224456336512, −2.84950437710011496969582801352, −2.00287272584340577275989446066, −1.75058722363837353002604825608, −1.70435719291961066059330028805, −0.996581946062204394663695197269, −0.17593646263194472129685502102, −0.15532174894196887778913313323,
0.15532174894196887778913313323, 0.17593646263194472129685502102, 0.996581946062204394663695197269, 1.70435719291961066059330028805, 1.75058722363837353002604825608, 2.00287272584340577275989446066, 2.84950437710011496969582801352, 2.94548301039826315224456336512, 4.00850385571617367165391393664, 4.26156220608959062701388963697, 4.45097609184465764700815762996, 4.63677026266559393030178963413, 5.25782778861953549805928775161, 5.57876702735004599115221210538, 5.84319643579404805015854289831, 6.54684725007864133721823545594, 6.83976838095530429117676897964, 6.97791762334062827919648444075, 7.53158066427751035955554867823, 7.65148445992319205146466931819, 7.959652229030541658060876250255, 8.113249293398086687134823507742, 8.733317417286022271261390996046, 8.923342062199955172726526362172, 8.989290538003359400676858255202
