| L(s) = 1 | − 9-s − 12·23-s − 25-s + 12·41-s − 12·47-s − 14·49-s − 24·71-s + 4·73-s + 81-s − 12·89-s + 20·97-s + 24·103-s + 24·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.50·23-s − 1/5·25-s + 1.87·41-s − 1.75·47-s − 2·49-s − 2.84·71-s + 0.468·73-s + 1/9·81-s − 1.27·89-s + 2.03·97-s + 2.36·103-s + 2.25·113-s + 2·121-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 + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.285646034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.285646034\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.21920542835559312492446111555, −9.793247901647739489520872979821, −9.577741797241966118999245445284, −8.824070133905395271991144826256, −8.705683063084889881576061197625, −7.961204248374949206670270006022, −7.87916085610923224447020596124, −7.42240725097047349228956579333, −6.80884305045386469560684467382, −6.18812499032682209660719361391, −6.05741640814622217771019205306, −5.62397999130265300735363697996, −4.92866893943400362426727922435, −4.37377298093735917814705833850, −4.15890611029263288064208198433, −3.21524134108264886005687320296, −3.10913751543731219744445176133, −1.98470210193189163206712026617, −1.83583404981582054596025117563, −0.51270590044254148579088651385,
0.51270590044254148579088651385, 1.83583404981582054596025117563, 1.98470210193189163206712026617, 3.10913751543731219744445176133, 3.21524134108264886005687320296, 4.15890611029263288064208198433, 4.37377298093735917814705833850, 4.92866893943400362426727922435, 5.62397999130265300735363697996, 6.05741640814622217771019205306, 6.18812499032682209660719361391, 6.80884305045386469560684467382, 7.42240725097047349228956579333, 7.87916085610923224447020596124, 7.961204248374949206670270006022, 8.705683063084889881576061197625, 8.824070133905395271991144826256, 9.577741797241966118999245445284, 9.793247901647739489520872979821, 10.21920542835559312492446111555
