| L(s) = 1 | + 4·4-s + 12·16-s + 2·19-s − 14·31-s − 11·49-s − 26·61-s + 32·64-s + 8·76-s + 8·79-s + 38·109-s − 22·121-s − 56·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3·16-s + 0.458·19-s − 2.51·31-s − 1.57·49-s − 3.32·61-s + 4·64-s + 0.917·76-s + 0.900·79-s + 3.63·109-s − 2·121-s − 5.02·124-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 + 1/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.190215000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.190215000\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 169 T^{2} + p^{2} T^{4} \) |
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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
−12.34436268877773826824390940586, −11.97213337598336306677428842685, −11.42765041955848287325130166705, −11.13023079679541338647056538270, −10.55139120736728572680468902968, −10.45407558964146273602629526461, −9.446957461320592505642302542068, −9.342256799032793945735448481535, −8.348809616381345941528354267912, −7.87110470045488970311903019173, −7.32422040193434457526132742217, −7.09361384705354003603112536606, −6.36362376933091395923522437303, −5.92646873641032548968684219242, −5.42886869452132221315686201870, −4.59633600622430725213246465689, −3.42938708516637227639041179406, −3.20639718690438586361714665153, −2.15579548274090720163004865639, −1.55320214804596082455230921611,
1.55320214804596082455230921611, 2.15579548274090720163004865639, 3.20639718690438586361714665153, 3.42938708516637227639041179406, 4.59633600622430725213246465689, 5.42886869452132221315686201870, 5.92646873641032548968684219242, 6.36362376933091395923522437303, 7.09361384705354003603112536606, 7.32422040193434457526132742217, 7.87110470045488970311903019173, 8.348809616381345941528354267912, 9.342256799032793945735448481535, 9.446957461320592505642302542068, 10.45407558964146273602629526461, 10.55139120736728572680468902968, 11.13023079679541338647056538270, 11.42765041955848287325130166705, 11.97213337598336306677428842685, 12.34436268877773826824390940586
