| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s + 11-s − 12-s − 4·13-s − 16-s + 2·17-s − 18-s − 2·19-s − 22-s + 2·23-s + 3·24-s − 5·25-s + 4·26-s + 27-s − 2·29-s − 8·31-s − 5·32-s + 33-s − 2·34-s − 36-s − 6·37-s + 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 1.10·13-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.458·19-s − 0.213·22-s + 0.417·23-s + 0.612·24-s − 25-s + 0.784·26-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.883·32-s + 0.174·33-s − 0.342·34-s − 1/6·36-s − 0.986·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148137 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148137 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7694895280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7694895280\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 67 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.27571977229115, −12.80494646429509, −12.64708020623945, −11.91361781894536, −11.38525999529000, −10.73644660860776, −10.43691663401831, −9.720449822053813, −9.503594825633976, −9.085501129521388, −8.626876317059256, −7.963468103929716, −7.619482975401559, −7.264820335822157, −6.683221556120393, −5.807389305249776, −5.467924434564504, −4.588189789263935, −4.455971566129002, −3.603023596371675, −3.256746533105341, −2.291363668694701, −1.877952203169992, −1.193861739982344, −0.2964417915368658,
0.2964417915368658, 1.193861739982344, 1.877952203169992, 2.291363668694701, 3.256746533105341, 3.603023596371675, 4.455971566129002, 4.588189789263935, 5.467924434564504, 5.807389305249776, 6.683221556120393, 7.264820335822157, 7.619482975401559, 7.963468103929716, 8.626876317059256, 9.085501129521388, 9.503594825633976, 9.720449822053813, 10.43691663401831, 10.73644660860776, 11.38525999529000, 11.91361781894536, 12.64708020623945, 12.80494646429509, 13.27571977229115
