| L(s) = 1 | − 3-s − 3·5-s − 7-s + 9-s + 3·13-s + 3·15-s − 4·17-s + 19-s + 21-s − 4·23-s + 4·25-s − 27-s + 3·29-s + 2·31-s + 3·35-s − 3·37-s − 3·39-s + 4·43-s − 3·45-s − 3·47-s + 49-s + 4·51-s − 10·53-s − 57-s − 3·59-s + 2·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.832·13-s + 0.774·15-s − 0.970·17-s + 0.229·19-s + 0.218·21-s − 0.834·23-s + 4/5·25-s − 0.192·27-s + 0.557·29-s + 0.359·31-s + 0.507·35-s − 0.493·37-s − 0.480·39-s + 0.609·43-s − 0.447·45-s − 0.437·47-s + 1/7·49-s + 0.560·51-s − 1.37·53-s − 0.132·57-s − 0.390·59-s + 0.256·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−15.33313498924091, −14.56057931606043, −13.87849590566788, −13.50493855030554, −12.77761034027900, −12.34658600917569, −11.90313130226026, −11.34065497210532, −10.97989295614509, −10.49089017976752, −9.780903198229255, −9.202441870498393, −8.516661600840557, −8.091579632012934, −7.543771143109508, −6.886323299265794, −6.368779403174509, −5.929662300152402, −4.998446059788740, −4.532682237329573, −3.861240537246370, −3.481090929799379, −2.626087906419354, −1.681084886201869, −0.7334016322994767, 0,
0.7334016322994767, 1.681084886201869, 2.626087906419354, 3.481090929799379, 3.861240537246370, 4.532682237329573, 4.998446059788740, 5.929662300152402, 6.368779403174509, 6.886323299265794, 7.543771143109508, 8.091579632012934, 8.516661600840557, 9.202441870498393, 9.780903198229255, 10.49089017976752, 10.97989295614509, 11.34065497210532, 11.90313130226026, 12.34658600917569, 12.77761034027900, 13.50493855030554, 13.87849590566788, 14.56057931606043, 15.33313498924091
