L(s) = 1 | − 2·3-s + 5-s + 6·7-s + 9-s − 11-s + 10·13-s − 2·15-s + 8·17-s − 5·19-s − 12·21-s + 8·23-s − 4·25-s + 2·27-s + 6·29-s − 2·31-s + 2·33-s + 6·35-s − 3·37-s − 20·39-s + 12·41-s + 14·43-s + 45-s − 12·47-s + 13·49-s − 16·51-s − 11·53-s − 55-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s + 2.26·7-s + 1/3·9-s − 0.301·11-s + 2.77·13-s − 0.516·15-s + 1.94·17-s − 1.14·19-s − 2.61·21-s + 1.66·23-s − 4/5·25-s + 0.384·27-s + 1.11·29-s − 0.359·31-s + 0.348·33-s + 1.01·35-s − 0.493·37-s − 3.20·39-s + 1.87·41-s + 2.13·43-s + 0.149·45-s − 1.75·47-s + 13/7·49-s − 2.24·51-s − 1.51·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.732498443\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.732498443\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + T - 7 T^{2} - 14 T^{3} - 68 T^{4} - 14 p T^{5} - 7 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 5 T - 5 T^{2} - 40 T^{3} + 64 T^{4} - 40 p T^{5} - 5 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 3 T - 53 T^{2} - 36 T^{3} + 2142 T^{4} - 36 p T^{5} - 53 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 11 T - T^{2} + 176 T^{3} + 5662 T^{4} + 176 p T^{5} - p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 5 T - 85 T^{2} + 40 T^{3} + 7144 T^{4} + 40 p T^{5} - 85 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 7 T - 83 T^{2} + 14 T^{3} + 9652 T^{4} + 14 p T^{5} - 83 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - T - 131 T^{2} + 14 T^{3} + 12022 T^{4} + 14 p T^{5} - 131 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 8 T - 53 T^{2} + 328 T^{3} + 2392 T^{4} + 328 p T^{5} - 53 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 7 T + 164 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T - 94 T^{2} + 288 T^{3} + 5775 T^{4} + 288 p T^{5} - 94 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 25 T + 336 T^{2} - 25 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.457186675557498050017527668041, −8.101159551573415746623319156499, −7.83008210436565779230548107869, −7.77676323199870867804779541776, −7.49059981103733252696925178363, −7.18681853954463542255624733924, −6.78154421933713806916810779462, −6.43003369457802944077591300045, −6.15204781159697356163589170827, −6.01533140034928252569747787192, −5.86279335594290059257680124061, −5.57128198500792941915531630452, −5.34384840303546732880094284646, −4.86776942251035515082418012651, −4.63217650865892246754143326071, −4.54885706073936444995144073983, −4.32635298884574786485131137720, −3.45923818352398015171569409888, −3.42452853812675334369098323117, −3.33819611119224192321665850834, −2.45882049048464737088415181622, −2.14477394519740374842223426050, −1.56565564908952444260223596235, −1.11471273770735904449800676994, −1.04862315373971995582105982313,
1.04862315373971995582105982313, 1.11471273770735904449800676994, 1.56565564908952444260223596235, 2.14477394519740374842223426050, 2.45882049048464737088415181622, 3.33819611119224192321665850834, 3.42452853812675334369098323117, 3.45923818352398015171569409888, 4.32635298884574786485131137720, 4.54885706073936444995144073983, 4.63217650865892246754143326071, 4.86776942251035515082418012651, 5.34384840303546732880094284646, 5.57128198500792941915531630452, 5.86279335594290059257680124061, 6.01533140034928252569747787192, 6.15204781159697356163589170827, 6.43003369457802944077591300045, 6.78154421933713806916810779462, 7.18681853954463542255624733924, 7.49059981103733252696925178363, 7.77676323199870867804779541776, 7.83008210436565779230548107869, 8.101159551573415746623319156499, 8.457186675557498050017527668041