| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 3·5-s − 4·6-s + 2·7-s + 4·8-s + 3·9-s + 6·10-s − 11-s − 6·12-s + 3·13-s + 4·14-s − 6·15-s + 5·16-s − 17-s + 6·18-s − 19-s + 9·20-s − 4·21-s − 2·22-s − 16·23-s − 8·24-s + 25-s + 6·26-s − 4·27-s + 6·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.34·5-s − 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s + 1.89·10-s − 0.301·11-s − 1.73·12-s + 0.832·13-s + 1.06·14-s − 1.54·15-s + 5/4·16-s − 0.242·17-s + 1.41·18-s − 0.229·19-s + 2.01·20-s − 0.872·21-s − 0.426·22-s − 3.33·23-s − 1.63·24-s + 1/5·25-s + 1.17·26-s − 0.769·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34596 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34596 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.668125374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.668125374\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 120 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 13 T + 138 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T + 150 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 126 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 5 T + 66 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 108 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−12.82970242883536565766697726296, −12.41632016184871497591587058850, −11.77635719893748808657083014272, −11.64292813372037331744539020433, −10.99724376699698323288834005611, −10.58074293327207340388459008405, −9.960164633792659717568248243671, −9.848792134156071187791953251953, −8.813422363596946949301763839220, −8.020998539187196701231981613739, −7.67343845024421837813346723312, −6.65478381999053354976868805730, −6.33537835797121285546506423012, −5.89527281517259554048250840687, −5.32146829320858463202368391376, −5.08763962406561887319709169070, −4.02831647294226359807140406812, −3.74481057082569996274433617289, −2.02882643892120175759435607569, −1.91959031414303079475692139098,
1.91959031414303079475692139098, 2.02882643892120175759435607569, 3.74481057082569996274433617289, 4.02831647294226359807140406812, 5.08763962406561887319709169070, 5.32146829320858463202368391376, 5.89527281517259554048250840687, 6.33537835797121285546506423012, 6.65478381999053354976868805730, 7.67343845024421837813346723312, 8.020998539187196701231981613739, 8.813422363596946949301763839220, 9.848792134156071187791953251953, 9.960164633792659717568248243671, 10.58074293327207340388459008405, 10.99724376699698323288834005611, 11.64292813372037331744539020433, 11.77635719893748808657083014272, 12.41632016184871497591587058850, 12.82970242883536565766697726296
