| L(s) = 1 | − 8·4-s + 13·7-s + 48·16-s + 50·25-s − 104·28-s − 94·37-s + 44·43-s + 120·49-s − 256·64-s − 218·67-s + 262·79-s − 400·100-s + 428·109-s + 624·112-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 752·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 337·169-s − 352·172-s + ⋯ |
| L(s) = 1 | − 2·4-s + 13/7·7-s + 3·16-s + 2·25-s − 3.71·28-s − 2.54·37-s + 1.02·43-s + 2.44·49-s − 4·64-s − 3.25·67-s + 3.31·79-s − 4·100-s + 3.92·109-s + 39/7·112-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 5.08·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.99·169-s − 2.04·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.409658043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.409658043\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - 13 T + p^{2} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )( 1 + T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 11 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 47 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 121 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 109 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 97 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 167 T + p^{2} T^{2} )( 1 + 167 T + p^{2} T^{2} ) \) |
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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
−12.39468578412390354196596857084, −12.36116892890423398699508567263, −11.71806896707312518314054765667, −10.93626709095079834815450191126, −10.49939031211302205708103797508, −10.31949002103609815991745590462, −9.345521451682793689348305944786, −8.966900277572951362112596132517, −8.634578279212685524118154523541, −8.241270827466330803904759730102, −7.60373479067498568667962273143, −7.16054145908955019311848036993, −6.07476919334701034285955961500, −5.38040556498843869904585049803, −4.84451595149901926247581337760, −4.71065469859008843809431811849, −3.91257805696893298484065793291, −3.14487885448521801597493251481, −1.72130211511069732702418021271, −0.78452085254629715719155717232,
0.78452085254629715719155717232, 1.72130211511069732702418021271, 3.14487885448521801597493251481, 3.91257805696893298484065793291, 4.71065469859008843809431811849, 4.84451595149901926247581337760, 5.38040556498843869904585049803, 6.07476919334701034285955961500, 7.16054145908955019311848036993, 7.60373479067498568667962273143, 8.241270827466330803904759730102, 8.634578279212685524118154523541, 8.966900277572951362112596132517, 9.345521451682793689348305944786, 10.31949002103609815991745590462, 10.49939031211302205708103797508, 10.93626709095079834815450191126, 11.71806896707312518314054765667, 12.36116892890423398699508567263, 12.39468578412390354196596857084
