| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 2·11-s + 5·16-s − 4·22-s − 12·23-s − 10·25-s + 8·29-s − 6·32-s + 4·37-s + 20·43-s + 6·44-s + 24·46-s + 20·50-s − 4·53-s − 16·58-s + 7·64-s + 16·67-s − 32·71-s − 8·74-s − 16·79-s − 40·86-s − 8·88-s − 36·92-s − 30·100-s + 8·106-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 0.603·11-s + 5/4·16-s − 0.852·22-s − 2.50·23-s − 2·25-s + 1.48·29-s − 1.06·32-s + 0.657·37-s + 3.04·43-s + 0.904·44-s + 3.53·46-s + 2.82·50-s − 0.549·53-s − 2.10·58-s + 7/8·64-s + 1.95·67-s − 3.79·71-s − 0.929·74-s − 1.80·79-s − 4.31·86-s − 0.852·88-s − 3.75·92-s − 3·100-s + 0.777·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
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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
−7.61306098166435554070562429733, −7.49151797748353724339772904966, −6.73434738171616199134482069220, −6.65384527303565678162733941289, −6.15284543364006056548718299211, −5.96843709024867243502708045318, −5.67611566136597490879348210320, −5.35891266185092378158829619906, −4.53764696532617191512962115334, −4.30847609020242236869298969714, −3.87591841174532161644364408016, −3.81771704539733165727045683571, −2.91948815150303705673915894821, −2.73373241935686791236895631579, −2.19371925871591313739375559171, −1.96006000917457453118379825711, −1.22292499567795485421307076375, −1.10508380038814680699142894786, 0, 0,
1.10508380038814680699142894786, 1.22292499567795485421307076375, 1.96006000917457453118379825711, 2.19371925871591313739375559171, 2.73373241935686791236895631579, 2.91948815150303705673915894821, 3.81771704539733165727045683571, 3.87591841174532161644364408016, 4.30847609020242236869298969714, 4.53764696532617191512962115334, 5.35891266185092378158829619906, 5.67611566136597490879348210320, 5.96843709024867243502708045318, 6.15284543364006056548718299211, 6.65384527303565678162733941289, 6.73434738171616199134482069220, 7.49151797748353724339772904966, 7.61306098166435554070562429733
