| L(s) = 1 | − 2·3-s − 7-s + 9-s − 4·11-s − 4·13-s + 2·17-s + 6·19-s + 2·21-s − 8·23-s − 5·25-s + 4·27-s + 6·29-s − 8·31-s + 8·33-s + 37-s + 8·39-s − 2·41-s + 4·43-s + 4·47-s + 49-s − 4·51-s + 10·53-s − 12·57-s + 14·59-s − 63-s − 8·67-s + 16·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.485·17-s + 1.37·19-s + 0.436·21-s − 1.66·23-s − 25-s + 0.769·27-s + 1.11·29-s − 1.43·31-s + 1.39·33-s + 0.164·37-s + 1.28·39-s − 0.312·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.560·51-s + 1.37·53-s − 1.58·57-s + 1.82·59-s − 0.125·63-s − 0.977·67-s + 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−16.10598808905931, −15.92979706276107, −15.29183125519010, −14.39679857508293, −14.04355298349890, −13.37878977952635, −12.63186242680066, −12.27413806000717, −11.71584093189063, −11.34433625321861, −10.43472634160433, −10.04506727881650, −9.775228518176262, −8.788216579007425, −8.035443113969632, −7.422919113504692, −7.044675174515023, −6.026832437759474, −5.603115031006940, −5.255523111210486, −4.458785029004850, −3.624989875209030, −2.752980768994085, −2.075709424503253, −0.7725532117512685, 0,
0.7725532117512685, 2.075709424503253, 2.752980768994085, 3.624989875209030, 4.458785029004850, 5.255523111210486, 5.603115031006940, 6.026832437759474, 7.044675174515023, 7.422919113504692, 8.035443113969632, 8.788216579007425, 9.775228518176262, 10.04506727881650, 10.43472634160433, 11.34433625321861, 11.71584093189063, 12.27413806000717, 12.63186242680066, 13.37878977952635, 14.04355298349890, 14.39679857508293, 15.29183125519010, 15.92979706276107, 16.10598808905931
