| L(s) = 1 | + 4-s − 3·11-s + 16-s − 12·23-s − 10·25-s − 8·31-s − 8·37-s − 3·44-s − 12·47-s − 10·49-s + 24·53-s + 6·59-s + 64-s + 10·67-s − 24·71-s + 12·89-s − 12·92-s + 10·97-s − 10·100-s + 28·103-s + 12·113-s − 2·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.904·11-s + 1/4·16-s − 2.50·23-s − 2·25-s − 1.43·31-s − 1.31·37-s − 0.452·44-s − 1.75·47-s − 1.42·49-s + 3.29·53-s + 0.781·59-s + 1/8·64-s + 1.22·67-s − 2.84·71-s + 1.27·89-s − 1.25·92-s + 1.01·97-s − 100-s + 2.75·103-s + 1.12·113-s − 0.181·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8209945948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8209945948\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−7.63671497466111343831458484773, −7.30919610404169303675315624830, −6.62996720332848440299588460341, −6.25936593777468405983226275094, −5.83411783800217827742089057037, −5.52999283985568650694698601644, −5.13805326110939411578475614907, −4.51360045423654490309797152466, −3.94847330019587176084970982031, −3.60725443430893185618920679500, −3.21966752389421416213948087838, −2.19879385611801234222671009191, −2.16127223562129545372435529545, −1.61170673287012800495909518619, −0.31179049021967099217986426436,
0.31179049021967099217986426436, 1.61170673287012800495909518619, 2.16127223562129545372435529545, 2.19879385611801234222671009191, 3.21966752389421416213948087838, 3.60725443430893185618920679500, 3.94847330019587176084970982031, 4.51360045423654490309797152466, 5.13805326110939411578475614907, 5.52999283985568650694698601644, 5.83411783800217827742089057037, 6.25936593777468405983226275094, 6.62996720332848440299588460341, 7.30919610404169303675315624830, 7.63671497466111343831458484773
