| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s − 2·9-s + 4·11-s − 12-s − 4·13-s + 16-s + 2·17-s + 2·18-s − 6·19-s − 4·22-s − 23-s + 24-s + 4·26-s + 5·27-s + 3·29-s + 4·31-s − 32-s − 4·33-s − 2·34-s − 2·36-s + 6·37-s + 6·38-s + 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s − 2/3·9-s + 1.20·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.485·17-s + 0.471·18-s − 1.37·19-s − 0.852·22-s − 0.208·23-s + 0.204·24-s + 0.784·26-s + 0.962·27-s + 0.557·29-s + 0.718·31-s − 0.176·32-s − 0.696·33-s − 0.342·34-s − 1/3·36-s + 0.986·37-s + 0.973·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.031835743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.031835743\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.89404420754535, −12.36632489190472, −12.06170342088449, −11.74308191736175, −11.16386160838571, −10.74601781777901, −10.30476390800260, −9.752026967714186, −9.329933401643466, −8.878915188576678, −8.348149962873069, −7.902951929723983, −7.381968570863459, −6.736961048721282, −6.226379022704584, −6.153910894723266, −5.335336448216692, −4.713057756641870, −4.333409546340581, −3.578037269485603, −2.905009243086693, −2.387132948111352, −1.760663689809965, −0.9605338270063804, −0.4003035642041004,
0.4003035642041004, 0.9605338270063804, 1.760663689809965, 2.387132948111352, 2.905009243086693, 3.578037269485603, 4.333409546340581, 4.713057756641870, 5.335336448216692, 6.153910894723266, 6.226379022704584, 6.736961048721282, 7.381968570863459, 7.902951929723983, 8.348149962873069, 8.878915188576678, 9.329933401643466, 9.752026967714186, 10.30476390800260, 10.74601781777901, 11.16386160838571, 11.74308191736175, 12.06170342088449, 12.36632489190472, 12.89404420754535
