| L(s) = 1 | − 7-s + 4·11-s − 13-s + 2·17-s + 4·19-s − 6·29-s − 8·31-s + 10·37-s + 6·41-s + 4·43-s − 8·47-s + 49-s + 6·53-s − 12·59-s − 2·61-s + 12·67-s + 8·71-s − 10·73-s − 4·77-s + 8·79-s + 4·83-s + 6·89-s + 91-s − 18·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.20·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s − 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.937·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s + 1.46·67-s + 0.949·71-s − 1.17·73-s − 0.455·77-s + 0.900·79-s + 0.439·83-s + 0.635·89-s + 0.104·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 163800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.45402077762347, −12.97725231050453, −12.54515287104034, −12.08814379377987, −11.60834850236139, −11.11290261159696, −10.80906029983924, −10.01729056992147, −9.438915141959425, −9.384674172605664, −8.927165577258299, −7.944657683181972, −7.806099333228276, −7.130145385924475, −6.740406374414319, −6.049379637018529, −5.730184150093694, −5.127130951693237, −4.487154287192477, −3.790588843185265, −3.592686905533513, −2.822017869190877, −2.205658840521406, −1.431895810053219, −0.9215479969399219, 0,
0.9215479969399219, 1.431895810053219, 2.205658840521406, 2.822017869190877, 3.592686905533513, 3.790588843185265, 4.487154287192477, 5.127130951693237, 5.730184150093694, 6.049379637018529, 6.740406374414319, 7.130145385924475, 7.806099333228276, 7.944657683181972, 8.927165577258299, 9.384674172605664, 9.438915141959425, 10.01729056992147, 10.80906029983924, 11.11290261159696, 11.60834850236139, 12.08814379377987, 12.54515287104034, 12.97725231050453, 13.45402077762347
