| L(s) = 1 | − 3·5-s + 7-s + 9-s − 11·11-s + 17-s + 2·23-s + 6·25-s + 4·27-s − 5·29-s − 3·31-s − 3·35-s − 3·37-s − 9·41-s + 11·43-s − 3·45-s − 2·47-s − 12·49-s + 21·53-s + 33·55-s − 14·59-s + 7·61-s + 63-s − 2·67-s − 22·71-s − 16·73-s − 11·77-s + 26·79-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s + 1/3·9-s − 3.31·11-s + 0.242·17-s + 0.417·23-s + 6/5·25-s + 0.769·27-s − 0.928·29-s − 0.538·31-s − 0.507·35-s − 0.493·37-s − 1.40·41-s + 1.67·43-s − 0.447·45-s − 0.291·47-s − 1.71·49-s + 2.88·53-s + 4.44·55-s − 1.82·59-s + 0.896·61-s + 0.125·63-s − 0.244·67-s − 2.61·71-s − 1.87·73-s − 1.25·77-s + 2.92·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.084649201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.084649201\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 37 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T^{2} - 4 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - T + 13 T^{2} - 4 T^{3} + 13 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + p T + 61 T^{2} + 234 T^{3} + 61 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - T^{2} - 32 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + 39 T^{2} - 42 T^{3} + 39 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 47 T^{2} - 4 T^{3} + 47 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 21 T^{2} - 156 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 5 T + 71 T^{2} + 214 T^{3} + 71 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 21 T^{2} - 84 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 9 T + 83 T^{2} + 374 T^{3} + 83 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 11 T + 145 T^{2} - 866 T^{3} + 145 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 111 T^{2} + 132 T^{3} + 111 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 21 T + 239 T^{2} - 1910 T^{3} + 239 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 14 T + 211 T^{2} + 1572 T^{3} + 211 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 7 T + 175 T^{2} - 850 T^{3} + 175 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 171 T^{2} + 212 T^{3} + 171 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 22 T + 277 T^{2} + 2484 T^{3} + 277 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 16 T + 255 T^{2} + 2128 T^{3} + 255 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 26 T + 407 T^{2} - 4332 T^{3} + 407 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2 T + 91 T^{2} + 12 p T^{3} + 91 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 97 | $S_4\times C_2$ | \( 1 + 21 T + 371 T^{2} + 3758 T^{3} + 371 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.61552873696836340248691502387, −7.57268605922585521371378856757, −7.38662444324926802399908197144, −7.24453615851395603015126479886, −6.86071968817547910821879254241, −6.46801628707090951076647729235, −6.29729126556632529448683531508, −5.79331665913031198525451721955, −5.51724865811946861676360263970, −5.47004372377051382802112506760, −5.02602647991771363589310278098, −4.94484667962681739192639261692, −4.62596051002762559569143569225, −4.40148377243558386877260217612, −3.99414867640331656712284046560, −3.74315767723857054453348570553, −3.19691709010331543783900725482, −3.17344639664564008140864476561, −2.89886976444819861530483923934, −2.44703344706390752919430050567, −2.17254553428172673548696839710, −1.76792687656029780591772315617, −1.30296297123430177828568643884, −0.55472076720297396479597959243, −0.35000398300654225355479344215,
0.35000398300654225355479344215, 0.55472076720297396479597959243, 1.30296297123430177828568643884, 1.76792687656029780591772315617, 2.17254553428172673548696839710, 2.44703344706390752919430050567, 2.89886976444819861530483923934, 3.17344639664564008140864476561, 3.19691709010331543783900725482, 3.74315767723857054453348570553, 3.99414867640331656712284046560, 4.40148377243558386877260217612, 4.62596051002762559569143569225, 4.94484667962681739192639261692, 5.02602647991771363589310278098, 5.47004372377051382802112506760, 5.51724865811946861676360263970, 5.79331665913031198525451721955, 6.29729126556632529448683531508, 6.46801628707090951076647729235, 6.86071968817547910821879254241, 7.24453615851395603015126479886, 7.38662444324926802399908197144, 7.57268605922585521371378856757, 7.61552873696836340248691502387
