| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 6·11-s − 12-s + 4·13-s + 16-s − 18-s − 19-s − 6·22-s + 6·23-s + 24-s − 5·25-s − 4·26-s − 27-s + 6·29-s + 4·31-s − 32-s − 6·33-s + 36-s + 2·37-s + 38-s − 4·39-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s − 0.235·18-s − 0.229·19-s − 1.27·22-s + 1.25·23-s + 0.204·24-s − 25-s − 0.784·26-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.04·33-s + 1/6·36-s + 0.328·37-s + 0.162·38-s − 0.640·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.536065612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536065612\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−8.448413415952757823859276826032, −7.30170772144122876227302062553, −6.68774061006756086268358265358, −6.21971937300053445652525318521, −5.49217727786485288614307314105, −4.32566530925924446527156549366, −3.80590551286803466423516961593, −2.68981811488085604602993737662, −1.43164106767407604874442455433, −0.871617597486924605171670414684,
0.871617597486924605171670414684, 1.43164106767407604874442455433, 2.68981811488085604602993737662, 3.80590551286803466423516961593, 4.32566530925924446527156549366, 5.49217727786485288614307314105, 6.21971937300053445652525318521, 6.68774061006756086268358265358, 7.30170772144122876227302062553, 8.448413415952757823859276826032
