| L(s) = 1 | + 4·7-s − 8·17-s − 8·23-s − 25-s + 8·31-s − 12·41-s − 16·47-s − 2·49-s − 16·71-s − 12·73-s + 8·79-s + 12·89-s − 28·97-s − 28·103-s − 32·113-s − 32·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 32·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.94·17-s − 1.66·23-s − 1/5·25-s + 1.43·31-s − 1.87·41-s − 2.33·47-s − 2/7·49-s − 1.89·71-s − 1.40·73-s + 0.900·79-s + 1.27·89-s − 2.84·97-s − 2.75·103-s − 3.01·113-s − 2.93·119-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 2.52·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33177600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33177600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4149031104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4149031104\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.214237187798307473687710618935, −8.082400256523640018282581260327, −7.898336721004458663399858952717, −7.00292078738893923101586874100, −6.93142814549780539705180739273, −6.62110122072285913163900606985, −6.20615840572147053438454479915, −5.72619023313681491768728941686, −5.45883953832016902731987089983, −4.85240049707536555008275018235, −4.67909559982567454125530140772, −4.26960590191886573809103148557, −4.19103009978474118572067126364, −3.32090664175627645551676191427, −3.12591327207140560339997469979, −2.40616837131909052644371411417, −2.05179052624208101002207894400, −1.59956901693486321499320967536, −1.33931723470777923243463564057, −0.15633567195328539711829573712,
0.15633567195328539711829573712, 1.33931723470777923243463564057, 1.59956901693486321499320967536, 2.05179052624208101002207894400, 2.40616837131909052644371411417, 3.12591327207140560339997469979, 3.32090664175627645551676191427, 4.19103009978474118572067126364, 4.26960590191886573809103148557, 4.67909559982567454125530140772, 4.85240049707536555008275018235, 5.45883953832016902731987089983, 5.72619023313681491768728941686, 6.20615840572147053438454479915, 6.62110122072285913163900606985, 6.93142814549780539705180739273, 7.00292078738893923101586874100, 7.898336721004458663399858952717, 8.082400256523640018282581260327, 8.214237187798307473687710618935
