| L(s) = 1 | + 5-s + 11-s + 3·13-s − 17-s − 2·19-s + 4·23-s − 4·25-s + 2·31-s − 7·37-s + 11·43-s + 10·47-s − 9·53-s + 55-s − 4·59-s − 4·61-s + 3·65-s + 13·67-s − 8·71-s + 73-s + 79-s − 9·83-s − 85-s + 3·89-s − 2·95-s − 7·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s + 0.832·13-s − 0.242·17-s − 0.458·19-s + 0.834·23-s − 4/5·25-s + 0.359·31-s − 1.15·37-s + 1.67·43-s + 1.45·47-s − 1.23·53-s + 0.134·55-s − 0.520·59-s − 0.512·61-s + 0.372·65-s + 1.58·67-s − 0.949·71-s + 0.117·73-s + 0.112·79-s − 0.987·83-s − 0.108·85-s + 0.317·89-s − 0.205·95-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.74822335732440, −13.45554598128202, −12.84012985454486, −12.35838728262791, −11.99940536500509, −11.21857998996112, −10.89322492472004, −10.57513441651374, −9.803710458891166, −9.436988888014767, −8.894679077743324, −8.517756895246652, −7.922330155911926, −7.304291495123132, −6.818260296572755, −6.247277705584857, −5.821093393664733, −5.333447500541082, −4.589987994785980, −4.094000737643677, −3.555937521081718, −2.840803245293572, −2.258145034409541, −1.546338209324946, −0.9827345466318640, 0,
0.9827345466318640, 1.546338209324946, 2.258145034409541, 2.840803245293572, 3.555937521081718, 4.094000737643677, 4.589987994785980, 5.333447500541082, 5.821093393664733, 6.247277705584857, 6.818260296572755, 7.304291495123132, 7.922330155911926, 8.517756895246652, 8.894679077743324, 9.436988888014767, 9.803710458891166, 10.57513441651374, 10.89322492472004, 11.21857998996112, 11.99940536500509, 12.35838728262791, 12.84012985454486, 13.45554598128202, 13.74822335732440
