| L(s) = 1 | − 2·4-s − 8·13-s + 4·16-s + 8·25-s + 4·37-s + 14·49-s + 16·52-s − 20·61-s − 8·64-s − 32·73-s + 16·97-s − 16·100-s + 40·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 4-s − 2.21·13-s + 16-s + 8/5·25-s + 0.657·37-s + 2·49-s + 2.21·52-s − 2.56·61-s − 64-s − 3.74·73-s + 1.62·97-s − 8/5·100-s + 3.83·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4739319503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4739319503\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−16.87136183962286060548292253966, −16.52648936978600694773576605124, −15.50835549989476606137386753619, −14.80336869328748637992644873152, −14.60737384875976943843829185846, −13.92483647868638341386564708123, −13.25198629763988668194638008179, −12.54150389996348553379979819090, −12.26445635236430275399076339267, −11.44800817633426214628243650161, −10.27676678139585969768072929358, −10.18182168954013904720347049411, −9.103835090281441945583940890664, −8.855499245950922138095638970351, −7.64417032236982256320859363833, −7.26184237017286218553976308465, −5.98621674530661777629858810811, −4.97377548933302703236598212025, −4.43545401389281113981475638487, −2.88466240953240789052920623905,
2.88466240953240789052920623905, 4.43545401389281113981475638487, 4.97377548933302703236598212025, 5.98621674530661777629858810811, 7.26184237017286218553976308465, 7.64417032236982256320859363833, 8.855499245950922138095638970351, 9.103835090281441945583940890664, 10.18182168954013904720347049411, 10.27676678139585969768072929358, 11.44800817633426214628243650161, 12.26445635236430275399076339267, 12.54150389996348553379979819090, 13.25198629763988668194638008179, 13.92483647868638341386564708123, 14.60737384875976943843829185846, 14.80336869328748637992644873152, 15.50835549989476606137386753619, 16.52648936978600694773576605124, 16.87136183962286060548292253966
