| L(s) = 1 | + 7-s − 8·19-s − 5·25-s + 7·31-s − 10·37-s + 13·43-s − 6·49-s − 13·61-s − 11·67-s + 17·73-s + 13·79-s + 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.83·19-s − 25-s + 1.25·31-s − 1.64·37-s + 1.98·43-s − 6/7·49-s − 1.66·61-s − 1.34·67-s + 1.98·73-s + 1.46·79-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.660354152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.660354152\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 13 T + p T^{2} \) | 1.43.an |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 17 T + p T^{2} \) | 1.73.ar |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.47259797451856, −14.91871193562321, −14.28151704726516, −13.83805893103300, −13.32464658175720, −12.61606779260195, −12.19840388567297, −11.69391056527222, −10.84690840104102, −10.66249862208934, −10.01399816436964, −9.213446347771120, −8.860431831495459, −7.982110267548641, −7.874398691201396, −6.880854126931617, −6.375711729957450, −5.833127294967698, −5.034271408174047, −4.415006176975455, −3.907414347580593, −3.033445644279319, −2.210990909270730, −1.647735341763304, −0.5015616455967059,
0.5015616455967059, 1.647735341763304, 2.210990909270730, 3.033445644279319, 3.907414347580593, 4.415006176975455, 5.034271408174047, 5.833127294967698, 6.375711729957450, 6.880854126931617, 7.874398691201396, 7.982110267548641, 8.860431831495459, 9.213446347771120, 10.01399816436964, 10.66249862208934, 10.84690840104102, 11.69391056527222, 12.19840388567297, 12.61606779260195, 13.32464658175720, 13.83805893103300, 14.28151704726516, 14.91871193562321, 15.47259797451856
