| L(s) = 1 | − 11-s + 13-s − 4·17-s − 4·19-s − 6·29-s − 2·37-s + 2·41-s + 4·43-s + 12·47-s − 7·49-s + 10·53-s − 8·59-s + 6·61-s − 2·67-s − 10·73-s + 10·79-s + 6·83-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s + 0.277·13-s − 0.970·17-s − 0.917·19-s − 1.11·29-s − 0.328·37-s + 0.312·41-s + 0.609·43-s + 1.75·47-s − 49-s + 1.37·53-s − 1.04·59-s + 0.768·61-s − 0.244·67-s − 1.17·73-s + 1.12·79-s + 0.658·83-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.46229982168805, −13.43300973798522, −12.79029905629393, −12.35019518794668, −11.83266574901098, −11.21938069917714, −10.83301653630736, −10.48928182925139, −9.906063520678862, −9.145734010097009, −8.988099372485711, −8.420343819361329, −7.786703069607118, −7.372850835407893, −6.774465263850420, −6.276069721508203, −5.753348439251508, −5.235230207322398, −4.545066936026939, −4.077154275324348, −3.594020495054251, −2.758004904005645, −2.245034310660496, −1.696994082124980, −0.7635646126831167, 0,
0.7635646126831167, 1.696994082124980, 2.245034310660496, 2.758004904005645, 3.594020495054251, 4.077154275324348, 4.545066936026939, 5.235230207322398, 5.753348439251508, 6.276069721508203, 6.774465263850420, 7.372850835407893, 7.786703069607118, 8.420343819361329, 8.988099372485711, 9.145734010097009, 9.906063520678862, 10.48928182925139, 10.83301653630736, 11.21938069917714, 11.83266574901098, 12.35019518794668, 12.79029905629393, 13.43300973798522, 13.46229982168805
