| L(s) = 1 | − 2·4-s + 2·7-s + 2·13-s + 4·16-s + 2·25-s − 4·28-s + 2·31-s + 14·37-s + 2·49-s − 4·52-s − 10·61-s − 8·64-s + 16·67-s − 22·79-s + 4·91-s − 4·100-s + 101-s + 103-s + 107-s + 109-s + 8·112-s + 113-s + 2·121-s − 4·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s + 0.755·7-s + 0.554·13-s + 16-s + 2/5·25-s − 0.755·28-s + 0.359·31-s + 2.30·37-s + 2/7·49-s − 0.554·52-s − 1.28·61-s − 64-s + 1.95·67-s − 2.47·79-s + 0.419·91-s − 2/5·100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.755·112-s + 0.0940·113-s + 2/11·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.033342181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.033342181\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−15.6573552799, −15.0493121231, −14.7989083334, −14.1265649677, −13.9792602007, −13.3186411952, −12.8699612441, −12.5414157211, −11.7734084601, −11.3403320356, −10.8796379876, −10.2435328648, −9.72560491070, −9.21164953256, −8.64977247819, −8.15104498432, −7.75993010545, −6.99953147484, −6.14051787937, −5.66287191424, −4.84339700748, −4.40255307769, −3.66989511421, −2.65205799832, −1.21843548350,
1.21843548350, 2.65205799832, 3.66989511421, 4.40255307769, 4.84339700748, 5.66287191424, 6.14051787937, 6.99953147484, 7.75993010545, 8.15104498432, 8.64977247819, 9.21164953256, 9.72560491070, 10.2435328648, 10.8796379876, 11.3403320356, 11.7734084601, 12.5414157211, 12.8699612441, 13.3186411952, 13.9792602007, 14.1265649677, 14.7989083334, 15.0493121231, 15.6573552799
