| L(s) = 1 | + 3·4-s + 5·16-s − 10·25-s + 12·37-s + 24·43-s + 3·64-s + 8·67-s + 16·79-s − 30·100-s − 36·109-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 36·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 72·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 5/4·16-s − 2·25-s + 1.97·37-s + 3.65·43-s + 3/8·64-s + 0.977·67-s + 1.80·79-s − 3·100-s − 3.44·109-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.95·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 5.48·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.491796100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.491796100\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−11.17895832286995930823270705118, −10.87499971733367981585631075971, −10.82216115514045736533181030394, −9.957838963733577647122838300373, −9.535213828304778577885788498129, −9.350293495462748629510472403158, −8.520751961180749240367235472642, −7.87819863379023890748178489750, −7.72390490908741287354403524651, −7.24600568074370316077773975368, −6.70112406592555219308003279582, −6.02340198806629958935966074475, −5.97076295085884490558763280650, −5.34158610926627434786098273374, −4.36693795227291559219296206399, −4.00801398992300897065524229330, −3.21673288771012501952651827597, −2.38459610907231863518447469134, −2.20191138215244987282741431171, −1.04262763156758973621639725624,
1.04262763156758973621639725624, 2.20191138215244987282741431171, 2.38459610907231863518447469134, 3.21673288771012501952651827597, 4.00801398992300897065524229330, 4.36693795227291559219296206399, 5.34158610926627434786098273374, 5.97076295085884490558763280650, 6.02340198806629958935966074475, 6.70112406592555219308003279582, 7.24600568074370316077773975368, 7.72390490908741287354403524651, 7.87819863379023890748178489750, 8.520751961180749240367235472642, 9.350293495462748629510472403158, 9.535213828304778577885788498129, 9.957838963733577647122838300373, 10.82216115514045736533181030394, 10.87499971733367981585631075971, 11.17895832286995930823270705118
