| L(s) = 1 | − 3·3-s − 3·5-s + 6·9-s − 5·11-s + 9·15-s + 17-s + 7·19-s + 23-s + 4·25-s − 9·27-s − 2·29-s + 5·31-s + 15·33-s + 11·37-s − 6·41-s + 12·43-s − 18·45-s + 7·47-s − 3·51-s + 11·53-s + 15·55-s − 21·57-s − 3·59-s + 5·61-s − 67-s − 3·69-s − 11·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s + 2·9-s − 1.50·11-s + 2.32·15-s + 0.242·17-s + 1.60·19-s + 0.208·23-s + 4/5·25-s − 1.73·27-s − 0.371·29-s + 0.898·31-s + 2.61·33-s + 1.80·37-s − 0.937·41-s + 1.82·43-s − 2.68·45-s + 1.02·47-s − 0.420·51-s + 1.51·53-s + 2.02·55-s − 2.78·57-s − 0.390·59-s + 0.640·61-s − 0.122·67-s − 0.361·69-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.51362804826923, −13.04968510843745, −12.60374167035248, −11.96884131176762, −11.92287042321430, −11.20938083976643, −11.11264999751832, −10.39792795964453, −10.10331165895932, −9.537530777511888, −8.791141192857409, −8.100427911917014, −7.562421837459509, −7.414415308813100, −6.900585946123381, −6.001290211678413, −5.729650798303675, −5.231911137569394, −4.643389713110555, −4.310975174085230, −3.590728259506125, −2.937667701739510, −2.277702484922133, −1.007956022344481, −0.7751149281360981, 0,
0.7751149281360981, 1.007956022344481, 2.277702484922133, 2.937667701739510, 3.590728259506125, 4.310975174085230, 4.643389713110555, 5.231911137569394, 5.729650798303675, 6.001290211678413, 6.900585946123381, 7.414415308813100, 7.562421837459509, 8.100427911917014, 8.791141192857409, 9.537530777511888, 10.10331165895932, 10.39792795964453, 11.11264999751832, 11.20938083976643, 11.92287042321430, 11.96884131176762, 12.60374167035248, 13.04968510843745, 13.51362804826923
