| L(s) = 1 | − 2·3-s − 4·5-s + 7-s + 9-s + 8·15-s − 2·19-s − 2·21-s + 8·23-s + 11·25-s + 4·27-s + 2·29-s + 4·31-s − 4·35-s − 6·37-s + 2·41-s + 8·43-s − 4·45-s + 4·47-s + 49-s + 10·53-s + 4·57-s + 6·59-s + 4·61-s + 63-s − 12·67-s − 16·69-s + 14·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s + 2.06·15-s − 0.458·19-s − 0.436·21-s + 1.66·23-s + 11/5·25-s + 0.769·27-s + 0.371·29-s + 0.718·31-s − 0.676·35-s − 0.986·37-s + 0.312·41-s + 1.21·43-s − 0.596·45-s + 0.583·47-s + 1/7·49-s + 1.37·53-s + 0.529·57-s + 0.781·59-s + 0.512·61-s + 0.125·63-s − 1.46·67-s − 1.92·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−13.71920878208695, −12.99422333516977, −12.57266882180038, −12.17330383601314, −11.69947732882869, −11.43433829604844, −10.96339125837473, −10.53293829657782, −10.21985447042045, −9.144000131060344, −8.750458816509189, −8.414326407501490, −7.647584661119117, −7.376378320895559, −6.780520262771040, −6.389797385166510, −5.622412476759347, −5.061331239921265, −4.731029381148366, −4.133721172758775, −3.658643010499106, −2.938206062628306, −2.353795746838987, −1.080970670506214, −0.7889968056911154, 0,
0.7889968056911154, 1.080970670506214, 2.353795746838987, 2.938206062628306, 3.658643010499106, 4.133721172758775, 4.731029381148366, 5.061331239921265, 5.622412476759347, 6.389797385166510, 6.780520262771040, 7.376378320895559, 7.647584661119117, 8.414326407501490, 8.750458816509189, 9.144000131060344, 10.21985447042045, 10.53293829657782, 10.96339125837473, 11.43433829604844, 11.69947732882869, 12.17330383601314, 12.57266882180038, 12.99422333516977, 13.71920878208695
