| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s − 4·11-s + 4·13-s − 2·14-s + 16-s − 4·19-s − 20-s − 4·22-s + 8·23-s + 25-s + 4·26-s − 2·28-s + 2·29-s − 4·31-s + 32-s + 2·35-s − 6·37-s − 4·38-s − 40-s + 8·41-s + 6·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s − 1.20·11-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s + 0.784·26-s − 0.377·28-s + 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.338·35-s − 0.986·37-s − 0.648·38-s − 0.158·40-s + 1.24·41-s + 0.914·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 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.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−15.71055800696071, −15.02993085729523, −14.63668545010703, −13.85810164864580, −13.28858316199856, −12.95385678575912, −12.59085232778409, −11.96458744912778, −10.99955533397514, −10.96733917408511, −10.41593902822206, −9.600524827552298, −8.902331752174900, −8.455500412504370, −7.681693605807159, −7.219824328886328, −6.476951557415782, −6.081389950925155, −5.306411176905996, −4.779145411493893, −4.046347249313745, −3.350227823358517, −2.932160858140018, −2.116164882993400, −1.056915372924925, 0,
1.056915372924925, 2.116164882993400, 2.932160858140018, 3.350227823358517, 4.046347249313745, 4.779145411493893, 5.306411176905996, 6.081389950925155, 6.476951557415782, 7.219824328886328, 7.681693605807159, 8.455500412504370, 8.902331752174900, 9.600524827552298, 10.41593902822206, 10.96733917408511, 10.99955533397514, 11.96458744912778, 12.59085232778409, 12.95385678575912, 13.28858316199856, 13.85810164864580, 14.63668545010703, 15.02993085729523, 15.71055800696071
