| L(s) = 1 | − 3-s − 4·7-s + 9-s + 2·11-s + 2·13-s + 2·17-s − 4·19-s + 4·21-s − 23-s − 5·25-s − 27-s − 2·29-s − 2·33-s + 6·37-s − 2·39-s + 41-s + 2·43-s + 12·47-s + 9·49-s − 2·51-s + 2·53-s + 4·57-s − 12·59-s + 14·61-s − 4·63-s − 4·67-s + 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.872·21-s − 0.208·23-s − 25-s − 0.192·27-s − 0.371·29-s − 0.348·33-s + 0.986·37-s − 0.320·39-s + 0.156·41-s + 0.304·43-s + 1.75·47-s + 9/7·49-s − 0.280·51-s + 0.274·53-s + 0.529·57-s − 1.56·59-s + 1.79·61-s − 0.503·63-s − 0.488·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 181056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 23 | \( 1 + T \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.33416944714914, −12.78980482343741, −12.54102329974722, −12.02827122897885, −11.52430145303379, −11.08695100174797, −10.44869213162384, −10.17291172288393, −9.573918452516455, −9.240624079921563, −8.763476852045695, −8.094170598602235, −7.505528851011542, −7.056615097596462, −6.424015270334607, −6.102102652668652, −5.824591826247140, −5.156735327121864, −4.296712189779563, −3.924043490603974, −3.562824654514917, −2.759440713939999, −2.231428645261055, −1.382758852272988, −0.6990517622483797, 0,
0.6990517622483797, 1.382758852272988, 2.231428645261055, 2.759440713939999, 3.562824654514917, 3.924043490603974, 4.296712189779563, 5.156735327121864, 5.824591826247140, 6.102102652668652, 6.424015270334607, 7.056615097596462, 7.505528851011542, 8.094170598602235, 8.763476852045695, 9.240624079921563, 9.573918452516455, 10.17291172288393, 10.44869213162384, 11.08695100174797, 11.52430145303379, 12.02827122897885, 12.54102329974722, 12.78980482343741, 13.33416944714914
