L(s) = 1 | + 4·5-s − 16·19-s + 11·25-s + 16·29-s − 8·31-s − 24·41-s − 49-s − 16·59-s − 12·61-s + 16·79-s − 8·89-s − 64·95-s − 24·101-s − 4·109-s − 22·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s − 32·155-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 3.67·19-s + 11/5·25-s + 2.97·29-s − 1.43·31-s − 3.74·41-s − 1/7·49-s − 2.08·59-s − 1.53·61-s + 1.80·79-s − 0.847·89-s − 6.56·95-s − 2.38·101-s − 0.383·109-s − 2·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.604675731\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.604675731\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.687034078389888599956455145137, −8.255991205840487765525451927045, −7.85003515183107985585570864504, −7.11649220416291560652323164152, −6.74201679680217856153555134743, −6.46979954200226927275397860038, −6.44348308624615071114584636469, −6.03382695490787859462403850567, −5.52558682272690086878815941010, −5.06971579066216886746493290452, −4.86064859011381207697476780291, −4.35364185073511004080108465126, −4.14882272281692249035156862655, −3.33516454100800292085131580258, −3.02303042816921714036285258224, −2.52539858277502070462823824253, −2.06551600039649436516806368818, −1.69386519481983963139667836046, −1.41317814317771354390157819295, −0.30863265922753188692794668314,
0.30863265922753188692794668314, 1.41317814317771354390157819295, 1.69386519481983963139667836046, 2.06551600039649436516806368818, 2.52539858277502070462823824253, 3.02303042816921714036285258224, 3.33516454100800292085131580258, 4.14882272281692249035156862655, 4.35364185073511004080108465126, 4.86064859011381207697476780291, 5.06971579066216886746493290452, 5.52558682272690086878815941010, 6.03382695490787859462403850567, 6.44348308624615071114584636469, 6.46979954200226927275397860038, 6.74201679680217856153555134743, 7.11649220416291560652323164152, 7.85003515183107985585570864504, 8.255991205840487765525451927045, 8.687034078389888599956455145137