| L(s) = 1 | − 2·3-s + 2·7-s + 2·9-s + 4·11-s − 6·13-s − 2·19-s − 4·21-s + 8·23-s − 6·27-s − 4·31-s − 8·33-s + 12·39-s − 8·41-s + 4·43-s + 12·47-s + 3·49-s + 20·53-s + 4·57-s + 14·59-s + 18·61-s + 4·63-s + 8·67-s − 16·69-s − 8·71-s − 12·73-s + 8·77-s − 8·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 2/3·9-s + 1.20·11-s − 1.66·13-s − 0.458·19-s − 0.872·21-s + 1.66·23-s − 1.15·27-s − 0.718·31-s − 1.39·33-s + 1.92·39-s − 1.24·41-s + 0.609·43-s + 1.75·47-s + 3/7·49-s + 2.74·53-s + 0.529·57-s + 1.82·59-s + 2.30·61-s + 0.503·63-s + 0.977·67-s − 1.92·69-s − 0.949·71-s − 1.40·73-s + 0.911·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.853282065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.853282065\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 162 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 198 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.494299681843990775754783993499, −7.86967594969865632046322553279, −7.29591350112067380320421947588, −7.18921458988216958824802347561, −6.91700962725597000632994447884, −6.80193677632362822862814000883, −6.02641101460420488033554385349, −5.71644742531830893508861749753, −5.33491829816260967127531176172, −5.27443686824406464644882257803, −4.80214469219073712678488825245, −4.19447431799492039169096817494, −4.01224227945188727011667102237, −3.80795064199861518719228141394, −2.74769569617473279197927234710, −2.70577648959348980990193180989, −1.98383871145232652840662099011, −1.59070357430795627043661876174, −0.900121281086606078950076911611, −0.48147935295119514820815725532,
0.48147935295119514820815725532, 0.900121281086606078950076911611, 1.59070357430795627043661876174, 1.98383871145232652840662099011, 2.70577648959348980990193180989, 2.74769569617473279197927234710, 3.80795064199861518719228141394, 4.01224227945188727011667102237, 4.19447431799492039169096817494, 4.80214469219073712678488825245, 5.27443686824406464644882257803, 5.33491829816260967127531176172, 5.71644742531830893508861749753, 6.02641101460420488033554385349, 6.80193677632362822862814000883, 6.91700962725597000632994447884, 7.18921458988216958824802347561, 7.29591350112067380320421947588, 7.86967594969865632046322553279, 8.494299681843990775754783993499
