| L(s) = 1 | + 2·3-s − 7-s + 9-s − 2·13-s + 4·19-s − 2·21-s − 2·25-s − 4·27-s − 8·31-s + 4·37-s − 4·39-s + 20·43-s + 49-s + 8·57-s + 22·61-s − 63-s − 4·67-s − 6·73-s − 4·75-s + 12·79-s − 11·81-s + 2·91-s − 16·93-s − 6·97-s + 8·103-s − 8·109-s + 8·111-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.917·19-s − 0.436·21-s − 2/5·25-s − 0.769·27-s − 1.43·31-s + 0.657·37-s − 0.640·39-s + 3.04·43-s + 1/7·49-s + 1.05·57-s + 2.81·61-s − 0.125·63-s − 0.488·67-s − 0.702·73-s − 0.461·75-s + 1.35·79-s − 1.22·81-s + 0.209·91-s − 1.65·93-s − 0.609·97-s + 0.788·103-s − 0.766·109-s + 0.759·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.598449015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.598449015\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.342379422854626178598697836373, −7.71020965535168762352447939797, −7.39607936959708834995065149212, −7.25764128203276037265040472238, −6.49968404699368296522321267019, −5.98412545990592658341511439264, −5.46992988264259474814177141652, −5.19691041320856891531216835201, −4.28485344699630660887490780153, −3.95118590009363919019527398227, −3.47361310427527045349746729129, −2.79239125658679579587797941223, −2.45987264487177963896762078217, −1.80203654184700977790344306125, −0.71968137914647596215738835629,
0.71968137914647596215738835629, 1.80203654184700977790344306125, 2.45987264487177963896762078217, 2.79239125658679579587797941223, 3.47361310427527045349746729129, 3.95118590009363919019527398227, 4.28485344699630660887490780153, 5.19691041320856891531216835201, 5.46992988264259474814177141652, 5.98412545990592658341511439264, 6.49968404699368296522321267019, 7.25764128203276037265040472238, 7.39607936959708834995065149212, 7.71020965535168762352447939797, 8.342379422854626178598697836373
