L(s) = 1 | + 3·5-s − 4·7-s + 13-s − 6·17-s + 8·19-s + 5·25-s − 9·29-s − 4·31-s − 12·35-s − 2·37-s − 6·41-s + 8·43-s − 12·47-s + 7·49-s − 12·53-s + 61-s + 3·65-s − 4·67-s + 24·71-s + 22·73-s − 16·79-s − 12·83-s − 18·85-s − 6·89-s − 4·91-s + 24·95-s − 2·97-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 1.51·7-s + 0.277·13-s − 1.45·17-s + 1.83·19-s + 25-s − 1.67·29-s − 0.718·31-s − 2.02·35-s − 0.328·37-s − 0.937·41-s + 1.21·43-s − 1.75·47-s + 49-s − 1.64·53-s + 0.128·61-s + 0.372·65-s − 0.488·67-s + 2.84·71-s + 2.57·73-s − 1.80·79-s − 1.31·83-s − 1.95·85-s − 0.635·89-s − 0.419·91-s + 2.46·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.696728008\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.696728008\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.740973106929054071644412810753, −9.583163462383997357265448249532, −9.363472196036309114401155880280, −8.606098134820989050472078579882, −8.532078949096744569256412001814, −7.72472787141542376709368112231, −7.28639871551875529939875308447, −6.85133063568742233099164183261, −6.61969604341525742722056257545, −5.99570089405431487669368317876, −5.90281503722265118375604612562, −5.18485958916772838256201943208, −5.05918822132392831946448573044, −4.21955633764197007030607844598, −3.61578157063667173648664509794, −3.23201982045174076559310766291, −2.79705550073498154863023221466, −1.96801266879127461765741723686, −1.68915131014969085525096244647, −0.52440637179999905284912982490,
0.52440637179999905284912982490, 1.68915131014969085525096244647, 1.96801266879127461765741723686, 2.79705550073498154863023221466, 3.23201982045174076559310766291, 3.61578157063667173648664509794, 4.21955633764197007030607844598, 5.05918822132392831946448573044, 5.18485958916772838256201943208, 5.90281503722265118375604612562, 5.99570089405431487669368317876, 6.61969604341525742722056257545, 6.85133063568742233099164183261, 7.28639871551875529939875308447, 7.72472787141542376709368112231, 8.532078949096744569256412001814, 8.606098134820989050472078579882, 9.363472196036309114401155880280, 9.583163462383997357265448249532, 9.740973106929054071644412810753