| L(s) = 1 | + 4-s + 9-s − 13-s + 16-s + 12·23-s + 2·25-s + 36-s − 4·43-s − 10·49-s − 52-s − 4·53-s + 12·61-s + 64-s + 4·79-s + 81-s + 12·92-s + 2·100-s + 20·101-s + 8·103-s + 4·107-s + 8·113-s − 117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s − 0.277·13-s + 1/4·16-s + 2.50·23-s + 2/5·25-s + 1/6·36-s − 0.609·43-s − 1.42·49-s − 0.138·52-s − 0.549·53-s + 1.53·61-s + 1/8·64-s + 0.450·79-s + 1/9·81-s + 1.25·92-s + 1/5·100-s + 1.99·101-s + 0.788·103-s + 0.386·107-s + 0.752·113-s − 0.0924·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79092 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79092 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.754466974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.754466974\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
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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
−9.858707796478671880273197037940, −9.152267755624316378786992783135, −8.864835874427244042022869682246, −8.232023786721150601266702548494, −7.62178752374908301327826244376, −7.21217125331025513970217224025, −6.62595094540068124903530115051, −6.38154612819467292375938844916, −5.39505991986435260793459916392, −5.03031301488734332531585164698, −4.45940338680708953947787054560, −3.47865313367211110836546013962, −3.02412058012420415522949932684, −2.15102200750335259550245751036, −1.11706595421072227978190780740,
1.11706595421072227978190780740, 2.15102200750335259550245751036, 3.02412058012420415522949932684, 3.47865313367211110836546013962, 4.45940338680708953947787054560, 5.03031301488734332531585164698, 5.39505991986435260793459916392, 6.38154612819467292375938844916, 6.62595094540068124903530115051, 7.21217125331025513970217224025, 7.62178752374908301327826244376, 8.232023786721150601266702548494, 8.864835874427244042022869682246, 9.152267755624316378786992783135, 9.858707796478671880273197037940
