| L(s) = 1 | + 5-s + 4·7-s + 11-s − 13-s − 2·17-s − 4·19-s + 25-s − 2·29-s − 8·31-s + 4·35-s − 6·37-s + 6·41-s + 4·43-s + 9·49-s + 2·53-s + 55-s − 12·59-s + 2·61-s − 65-s + 12·67-s − 14·73-s + 4·77-s + 4·83-s − 2·85-s + 10·89-s − 4·91-s − 4·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.301·11-s − 0.277·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 9/7·49-s + 0.274·53-s + 0.134·55-s − 1.56·59-s + 0.256·61-s − 0.124·65-s + 1.46·67-s − 1.63·73-s + 0.455·77-s + 0.439·83-s − 0.216·85-s + 1.05·89-s − 0.419·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−14.13429789000790, −13.48710574203706, −13.03452588610357, −12.45873660808924, −12.06879578614612, −11.40680264403984, −10.96995042411194, −10.72458387784185, −10.13628460697859, −9.356805936253329, −9.041589080030095, −8.549523403140021, −7.999537823214150, −7.464470330217877, −7.026221848586667, −6.353847005159391, −5.765909162812647, −5.282569448478397, −4.729044549988807, −4.238751126088677, −3.691659040742382, −2.807749784804541, −2.001430891855962, −1.852995790389896, −1.008945191479096, 0,
1.008945191479096, 1.852995790389896, 2.001430891855962, 2.807749784804541, 3.691659040742382, 4.238751126088677, 4.729044549988807, 5.282569448478397, 5.765909162812647, 6.353847005159391, 7.026221848586667, 7.464470330217877, 7.999537823214150, 8.549523403140021, 9.041589080030095, 9.356805936253329, 10.13628460697859, 10.72458387784185, 10.96995042411194, 11.40680264403984, 12.06879578614612, 12.45873660808924, 13.03452588610357, 13.48710574203706, 14.13429789000790
