| L(s) = 1 | + 5-s − 7-s − 6·11-s + 6·13-s + 2·17-s − 7·19-s − 23-s − 4·25-s − 2·29-s − 10·31-s − 35-s − 6·37-s + 8·41-s + 10·43-s + 8·47-s + 49-s − 2·53-s − 6·55-s + 7·61-s + 6·65-s + 12·67-s + 15·71-s − 2·73-s + 6·77-s − 79-s + 12·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.80·11-s + 1.66·13-s + 0.485·17-s − 1.60·19-s − 0.208·23-s − 4/5·25-s − 0.371·29-s − 1.79·31-s − 0.169·35-s − 0.986·37-s + 1.24·41-s + 1.52·43-s + 1.16·47-s + 1/7·49-s − 0.274·53-s − 0.809·55-s + 0.896·61-s + 0.744·65-s + 1.46·67-s + 1.78·71-s − 0.234·73-s + 0.683·77-s − 0.112·79-s + 1.31·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.569568710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.569568710\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.77763746173073565505191856517, −7.09780210344858379679174211826, −6.12547917369756228594767324529, −5.78112079418098408710951872504, −5.17853047863358314093283510722, −4.02645126327586566369449588302, −3.58191080352630978831036090478, −2.46517705035100110777270894962, −1.93590262537426860990021904555, −0.58056679883778899725336875125,
0.58056679883778899725336875125, 1.93590262537426860990021904555, 2.46517705035100110777270894962, 3.58191080352630978831036090478, 4.02645126327586566369449588302, 5.17853047863358314093283510722, 5.78112079418098408710951872504, 6.12547917369756228594767324529, 7.09780210344858379679174211826, 7.77763746173073565505191856517
