| L(s) = 1 | + 4-s + 8·5-s − 4·7-s − 3·9-s + 16-s − 4·17-s + 8·20-s + 38·25-s − 4·28-s − 32·35-s − 3·36-s − 8·37-s + 12·41-s + 20·43-s − 24·45-s + 9·49-s + 24·59-s + 12·63-s + 64-s − 20·67-s − 4·68-s − 24·79-s + 8·80-s + 9·81-s + 28·83-s − 32·85-s − 12·89-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 3.57·5-s − 1.51·7-s − 9-s + 1/4·16-s − 0.970·17-s + 1.78·20-s + 38/5·25-s − 0.755·28-s − 5.40·35-s − 1/2·36-s − 1.31·37-s + 1.87·41-s + 3.04·43-s − 3.57·45-s + 9/7·49-s + 3.12·59-s + 1.51·63-s + 1/8·64-s − 2.44·67-s − 0.485·68-s − 2.70·79-s + 0.894·80-s + 81-s + 3.07·83-s − 3.47·85-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.812652738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.812652738\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.473086403046087731503510874073, −7.49124187490201587466733057696, −6.98429022098037702749121152389, −6.69441913641732873591204403077, −6.24150742451512185812061050942, −5.84432417959727908710252168629, −5.73491872900764331079666777759, −5.42556424957105177850208163634, −4.65189030601306649410353288456, −3.90796663115617095498900880052, −2.98279695187217995742043408916, −2.63232737474552247848840602025, −2.38914481848313294289968516969, −1.80751679365655142616830080476, −0.897690950153135088654928702405,
0.897690950153135088654928702405, 1.80751679365655142616830080476, 2.38914481848313294289968516969, 2.63232737474552247848840602025, 2.98279695187217995742043408916, 3.90796663115617095498900880052, 4.65189030601306649410353288456, 5.42556424957105177850208163634, 5.73491872900764331079666777759, 5.84432417959727908710252168629, 6.24150742451512185812061050942, 6.69441913641732873591204403077, 6.98429022098037702749121152389, 7.49124187490201587466733057696, 8.473086403046087731503510874073
