| L(s) = 1 | − 3-s + 4·5-s − 2·7-s + 9-s + 11-s − 4·15-s + 6·17-s − 4·19-s + 2·21-s − 6·23-s + 11·25-s − 27-s − 33-s − 8·35-s − 6·37-s + 10·41-s + 8·43-s + 4·45-s − 6·47-s − 3·49-s − 6·51-s − 12·53-s + 4·55-s + 4·57-s − 8·59-s − 4·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.03·15-s + 1.45·17-s − 0.917·19-s + 0.436·21-s − 1.25·23-s + 11/5·25-s − 0.192·27-s − 0.174·33-s − 1.35·35-s − 0.986·37-s + 1.56·41-s + 1.21·43-s + 0.596·45-s − 0.875·47-s − 3/7·49-s − 0.840·51-s − 1.64·53-s + 0.539·55-s + 0.529·57-s − 1.04·59-s − 0.512·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.05671920042271, −12.77427532986350, −12.27961268955637, −12.07392440737460, −11.18544502838502, −10.71774918283423, −10.38264304657854, −9.895464813609090, −9.524951235850473, −9.241480909482403, −8.642334429337423, −7.854321317469621, −7.547026716946064, −6.678766473891626, −6.366751990534036, −5.970223245281479, −5.737538903152337, −5.051337115423983, −4.568948961534980, −3.860996604320716, −3.218026734009841, −2.705923828185115, −1.942385993081550, −1.592942715348314, −0.8729622036687484, 0,
0.8729622036687484, 1.592942715348314, 1.942385993081550, 2.705923828185115, 3.218026734009841, 3.860996604320716, 4.568948961534980, 5.051337115423983, 5.737538903152337, 5.970223245281479, 6.366751990534036, 6.678766473891626, 7.547026716946064, 7.854321317469621, 8.642334429337423, 9.241480909482403, 9.524951235850473, 9.895464813609090, 10.38264304657854, 10.71774918283423, 11.18544502838502, 12.07392440737460, 12.27961268955637, 12.77427532986350, 13.05671920042271
