| L(s) = 1 | − 2·3-s − 4-s − 3·9-s + 2·12-s − 3·16-s − 12·17-s − 7·25-s + 14·27-s + 6·29-s + 3·36-s − 22·43-s + 6·48-s + 24·51-s − 18·53-s + 14·61-s + 7·64-s + 12·68-s + 14·75-s − 10·79-s − 4·81-s − 12·87-s + 7·100-s + 18·101-s + 26·103-s − 14·108-s + 30·113-s − 6·116-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 9-s + 0.577·12-s − 3/4·16-s − 2.91·17-s − 7/5·25-s + 2.69·27-s + 1.11·29-s + 1/2·36-s − 3.35·43-s + 0.866·48-s + 3.36·51-s − 2.47·53-s + 1.79·61-s + 7/8·64-s + 1.45·68-s + 1.61·75-s − 1.12·79-s − 4/9·81-s − 1.28·87-s + 7/10·100-s + 1.79·101-s + 2.56·103-s − 1.34·108-s + 2.82·113-s − 0.557·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2529385137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2529385137\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 139 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 167 T^{2} + p^{2} T^{4} \) |
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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
−8.089799109950804278884308989466, −7.68346122861408504944010485183, −6.98156067914161030349707848378, −6.88262720664148651011220005946, −6.52880596323780344614386008529, −6.20590580112480446868341907673, −6.04579796176578473624398674480, −5.59480899171838539071068777479, −5.02201739826478910840454363774, −4.88202765744081354863580068994, −4.52795681251753775588740113069, −4.45285153901390918758731254088, −3.60066833940958077165977829105, −3.40793554288504661636708410561, −2.88745090190836110287968547029, −2.22943641110778235167606980813, −2.16357380472985321451759667851, −1.53455364214477769241479963296, −0.56158128140756504732569902067, −0.21985416398305173630138122930,
0.21985416398305173630138122930, 0.56158128140756504732569902067, 1.53455364214477769241479963296, 2.16357380472985321451759667851, 2.22943641110778235167606980813, 2.88745090190836110287968547029, 3.40793554288504661636708410561, 3.60066833940958077165977829105, 4.45285153901390918758731254088, 4.52795681251753775588740113069, 4.88202765744081354863580068994, 5.02201739826478910840454363774, 5.59480899171838539071068777479, 6.04579796176578473624398674480, 6.20590580112480446868341907673, 6.52880596323780344614386008529, 6.88262720664148651011220005946, 6.98156067914161030349707848378, 7.68346122861408504944010485183, 8.089799109950804278884308989466
