| L(s) = 1 | − 4-s − 9-s − 6·11-s + 16-s − 20·29-s + 2·31-s + 36-s + 14·41-s + 6·44-s + 5·49-s + 14·61-s − 64-s + 14·71-s + 20·79-s + 81-s − 20·89-s + 6·99-s + 4·101-s + 20·109-s + 20·116-s + 5·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 1.80·11-s + 1/4·16-s − 3.71·29-s + 0.359·31-s + 1/6·36-s + 2.18·41-s + 0.904·44-s + 5/7·49-s + 1.79·61-s − 1/8·64-s + 1.66·71-s + 2.25·79-s + 1/9·81-s − 2.11·89-s + 0.603·99-s + 0.398·101-s + 1.91·109-s + 1.85·116-s + 5/11·121-s − 0.179·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.237581977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.237581977\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.377604799309773937457146028845, −8.151355040010086038132006048414, −7.70324868955234278426824855259, −7.55702145021278217422385953648, −7.15585865794075265081974401189, −6.81281298641150295904781099619, −6.09173466326245110948278167203, −5.87919262427791740700407167773, −5.42204040940358163004452734879, −5.35998835911126706062063377877, −4.89721879316322916544077808694, −4.36759882774457445660590204323, −3.85595633741555357093632899715, −3.72878244180412921300063031332, −3.07884597862881339300639645792, −2.66174572948670480986817453271, −2.06560487194614340859647218523, −1.97313796200454235915259665725, −0.852049747919853985150696849212, −0.38749659757673643003144463942,
0.38749659757673643003144463942, 0.852049747919853985150696849212, 1.97313796200454235915259665725, 2.06560487194614340859647218523, 2.66174572948670480986817453271, 3.07884597862881339300639645792, 3.72878244180412921300063031332, 3.85595633741555357093632899715, 4.36759882774457445660590204323, 4.89721879316322916544077808694, 5.35998835911126706062063377877, 5.42204040940358163004452734879, 5.87919262427791740700407167773, 6.09173466326245110948278167203, 6.81281298641150295904781099619, 7.15585865794075265081974401189, 7.55702145021278217422385953648, 7.70324868955234278426824855259, 8.151355040010086038132006048414, 8.377604799309773937457146028845
