| L(s) = 1 | + 3·3-s − 6·5-s + 4·7-s + 6·9-s − 6·13-s − 18·15-s − 12·17-s − 19-s + 12·21-s + 19·25-s + 9·27-s + 6·29-s − 24·35-s − 18·39-s + 3·41-s + 8·43-s − 36·45-s − 6·47-s − 2·49-s − 36·51-s − 6·53-s − 3·57-s + 3·59-s − 10·61-s + 24·63-s + 36·65-s − 9·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 2.68·5-s + 1.51·7-s + 2·9-s − 1.66·13-s − 4.64·15-s − 2.91·17-s − 0.229·19-s + 2.61·21-s + 19/5·25-s + 1.73·27-s + 1.11·29-s − 4.05·35-s − 2.88·39-s + 0.468·41-s + 1.21·43-s − 5.36·45-s − 0.875·47-s − 2/7·49-s − 5.04·51-s − 0.824·53-s − 0.397·57-s + 0.390·59-s − 1.28·61-s + 3.02·63-s + 4.46·65-s − 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.673055844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.673055844\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 9 T + 94 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 5 T - 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 24 T + 271 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 139 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−10.41840626379780311296312983428, −9.661481057239537562704736764342, −9.287835119513140027982092698119, −8.824550681718801704146997609027, −8.478492103003179336485196609584, −8.194285299573093888085316549828, −7.895916634175371834038339563348, −7.43641298284835239214515304451, −7.30992682572251170484029342221, −6.78563242013722514051446588945, −6.24775450661600840688894080162, −4.96679150611870146457670861512, −4.57649365984010908194674420953, −4.45426573245439654641296831217, −4.27127357023424301750208968688, −3.31695438989022365791063079056, −3.10978147213474573631133134553, −2.08914749475677923902413133137, −2.08578192370435271478686706925, −0.53681737565738816201639661095,
0.53681737565738816201639661095, 2.08578192370435271478686706925, 2.08914749475677923902413133137, 3.10978147213474573631133134553, 3.31695438989022365791063079056, 4.27127357023424301750208968688, 4.45426573245439654641296831217, 4.57649365984010908194674420953, 4.96679150611870146457670861512, 6.24775450661600840688894080162, 6.78563242013722514051446588945, 7.30992682572251170484029342221, 7.43641298284835239214515304451, 7.895916634175371834038339563348, 8.194285299573093888085316549828, 8.478492103003179336485196609584, 8.824550681718801704146997609027, 9.287835119513140027982092698119, 9.661481057239537562704736764342, 10.41840626379780311296312983428
