| L(s) = 1 | − 2·4-s + 4·13-s + 4·16-s − 2·19-s − 9·23-s − 5·25-s + 9·29-s + 5·31-s − 2·37-s − 43-s + 9·47-s − 8·52-s − 9·59-s − 61-s − 8·64-s + 5·67-s + 9·71-s − 7·73-s + 4·76-s + 10·79-s + 9·89-s + 18·92-s − 97-s + 10·100-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 4-s + 1.10·13-s + 16-s − 0.458·19-s − 1.87·23-s − 25-s + 1.67·29-s + 0.898·31-s − 0.328·37-s − 0.152·43-s + 1.31·47-s − 1.10·52-s − 1.17·59-s − 0.128·61-s − 64-s + 0.610·67-s + 1.06·71-s − 0.819·73-s + 0.458·76-s + 1.12·79-s + 0.953·89-s + 1.87·92-s − 0.101·97-s + 100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 382347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 382347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−12.60563004081645, −12.26182289503336, −11.93626413343974, −11.41569117652543, −10.71814516153446, −10.41423289451848, −9.979551969842487, −9.589726107102383, −8.994110776835787, −8.589157716872438, −8.161614128490898, −7.916001617391737, −7.310705452152226, −6.468254209663803, −6.184161879869926, −5.857746281191817, −5.167286692053511, −4.666174881559257, −4.163400048933194, −3.810493851348530, −3.336344649794849, −2.574639536467593, −1.996097076302685, −1.284626055856103, −0.7135917339532115, 0,
0.7135917339532115, 1.284626055856103, 1.996097076302685, 2.574639536467593, 3.336344649794849, 3.810493851348530, 4.163400048933194, 4.666174881559257, 5.167286692053511, 5.857746281191817, 6.184161879869926, 6.468254209663803, 7.310705452152226, 7.916001617391737, 8.161614128490898, 8.589157716872438, 8.994110776835787, 9.589726107102383, 9.979551969842487, 10.41423289451848, 10.71814516153446, 11.41569117652543, 11.93626413343974, 12.26182289503336, 12.60563004081645
