L(s) = 1 | − 2·3-s + 3·9-s + 4·13-s − 4·17-s + 2·23-s + 25-s − 4·27-s + 10·29-s − 8·39-s − 18·43-s − 49-s + 8·51-s + 18·53-s − 16·61-s − 4·69-s − 2·75-s − 30·79-s + 5·81-s − 20·87-s − 16·101-s + 32·103-s + 24·107-s − 42·113-s + 12·117-s + 22·121-s + 127-s + 36·129-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s + 1.10·13-s − 0.970·17-s + 0.417·23-s + 1/5·25-s − 0.769·27-s + 1.85·29-s − 1.28·39-s − 2.74·43-s − 1/7·49-s + 1.12·51-s + 2.47·53-s − 2.04·61-s − 0.481·69-s − 0.230·75-s − 3.37·79-s + 5/9·81-s − 2.14·87-s − 1.59·101-s + 3.15·103-s + 2.32·107-s − 3.95·113-s + 1.10·117-s + 2·121-s + 0.0887·127-s + 3.16·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.446555214\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.446555214\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−8.483109430355364483188267080542, −8.322972365600663838213135272987, −7.85434813071926413684037945206, −7.22574316182143580436445420910, −6.98476355744776374355795453066, −6.72510900775202056675379231936, −6.37948599076330069735746101451, −5.98171745694268407693209292341, −5.65128778337520066821311069410, −5.31598700876528598401286302630, −4.68711036503520427731746008530, −4.55920942249123406172238277598, −4.22831860942247715270525487799, −3.61945291320940978966278365056, −3.09444572754715269326200259154, −2.85160880540150706177494747984, −1.98184444155926930509492809864, −1.62127115089319892757866895508, −1.00920474232791883544388227524, −0.43089322243059254102250581702,
0.43089322243059254102250581702, 1.00920474232791883544388227524, 1.62127115089319892757866895508, 1.98184444155926930509492809864, 2.85160880540150706177494747984, 3.09444572754715269326200259154, 3.61945291320940978966278365056, 4.22831860942247715270525487799, 4.55920942249123406172238277598, 4.68711036503520427731746008530, 5.31598700876528598401286302630, 5.65128778337520066821311069410, 5.98171745694268407693209292341, 6.37948599076330069735746101451, 6.72510900775202056675379231936, 6.98476355744776374355795453066, 7.22574316182143580436445420910, 7.85434813071926413684037945206, 8.322972365600663838213135272987, 8.483109430355364483188267080542