| L(s) = 1 | + 2·3-s + 9-s + 8·11-s − 4·13-s + 8·23-s + 25-s − 4·27-s + 16·33-s + 4·37-s − 8·39-s + 16·47-s − 49-s + 20·59-s − 20·61-s + 16·69-s + 4·71-s − 8·73-s + 2·75-s − 11·81-s − 4·83-s + 8·99-s − 8·107-s + 4·109-s + 8·111-s − 4·117-s + 30·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 2.41·11-s − 1.10·13-s + 1.66·23-s + 1/5·25-s − 0.769·27-s + 2.78·33-s + 0.657·37-s − 1.28·39-s + 2.33·47-s − 1/7·49-s + 2.60·59-s − 2.56·61-s + 1.92·69-s + 0.474·71-s − 0.936·73-s + 0.230·75-s − 1.22·81-s − 0.439·83-s + 0.804·99-s − 0.773·107-s + 0.383·109-s + 0.759·111-s − 0.369·117-s + 2.72·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.562033213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.562033213\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.620573831062798977338048306369, −7.74859565529322104393023388851, −7.51913798524960929657935182575, −6.94228855723361051473163441625, −6.77583442207330057269764252935, −6.10825255541183847603438633391, −5.63062233177260824192711786061, −4.99317742187020884036238960612, −4.39707270097592359909030964690, −4.00092343191497822357902611745, −3.53880448040920053957798177641, −2.85229035476262780880222464499, −2.48304539046153165764116554589, −1.64085529186781390840705154817, −0.942807742153316628183002052038,
0.942807742153316628183002052038, 1.64085529186781390840705154817, 2.48304539046153165764116554589, 2.85229035476262780880222464499, 3.53880448040920053957798177641, 4.00092343191497822357902611745, 4.39707270097592359909030964690, 4.99317742187020884036238960612, 5.63062233177260824192711786061, 6.10825255541183847603438633391, 6.77583442207330057269764252935, 6.94228855723361051473163441625, 7.51913798524960929657935182575, 7.74859565529322104393023388851, 8.620573831062798977338048306369
