| L(s) = 1 | − 3-s − 5-s + 9-s + 4·11-s − 13-s + 15-s − 6·17-s − 8·19-s − 2·23-s + 25-s − 27-s + 2·29-s − 4·33-s + 2·37-s + 39-s + 6·41-s − 45-s − 8·47-s + 6·51-s − 12·53-s − 4·55-s + 8·57-s + 4·59-s − 10·61-s + 65-s − 4·67-s + 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.258·15-s − 1.45·17-s − 1.83·19-s − 0.417·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.696·33-s + 0.328·37-s + 0.160·39-s + 0.937·41-s − 0.149·45-s − 1.16·47-s + 0.840·51-s − 1.64·53-s − 0.539·55-s + 1.05·57-s + 0.520·59-s − 1.28·61-s + 0.124·65-s − 0.488·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−13.40870859839387, −13.04586689900833, −12.45428767691643, −12.23998391016552, −11.53444844364195, −11.24306809056989, −10.81477440856565, −10.36154075894606, −9.699950578846787, −9.157166799039796, −8.852546245646444, −8.188654713793103, −7.806715683113841, −7.032883408650168, −6.576074658692868, −6.324261577016948, −5.848022370630910, −4.868583546575903, −4.531950313654657, −4.171320178751899, −3.590191762006196, −2.803002760048267, −2.049805844876666, −1.623814617154902, −0.6510386435810265, 0,
0.6510386435810265, 1.623814617154902, 2.049805844876666, 2.803002760048267, 3.590191762006196, 4.171320178751899, 4.531950313654657, 4.868583546575903, 5.848022370630910, 6.324261577016948, 6.576074658692868, 7.032883408650168, 7.806715683113841, 8.188654713793103, 8.852546245646444, 9.157166799039796, 9.699950578846787, 10.36154075894606, 10.81477440856565, 11.24306809056989, 11.53444844364195, 12.23998391016552, 12.45428767691643, 13.04586689900833, 13.40870859839387
