| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 6·11-s + 12-s − 14-s + 16-s + 6·17-s − 18-s + 6·19-s + 21-s − 6·22-s − 2·23-s − 24-s + 27-s + 28-s − 6·29-s + 4·31-s − 32-s + 6·33-s − 6·34-s + 36-s − 6·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 1.37·19-s + 0.218·21-s − 1.27·22-s − 0.417·23-s − 0.204·24-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.04·33-s − 1.02·34-s + 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.650059321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.650059321\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.30356727920397, −12.40794804091680, −12.11961506046011, −11.87640764749929, −11.27748668904297, −10.79740643112537, −10.16240598310771, −9.688276285021297, −9.367922660183149, −9.009237170440699, −8.392907632280261, −7.915821604353919, −7.437815907478363, −7.160893108142612, −6.326051253493476, −6.095864904371013, −5.291902919339613, −4.834925478639402, −4.009581568511514, −3.438615706901036, −3.289873476629441, −2.336622988148995, −1.586631810219707, −1.333678739213376, −0.6309191467529012,
0.6309191467529012, 1.333678739213376, 1.586631810219707, 2.336622988148995, 3.289873476629441, 3.438615706901036, 4.009581568511514, 4.834925478639402, 5.291902919339613, 6.095864904371013, 6.326051253493476, 7.160893108142612, 7.437815907478363, 7.915821604353919, 8.392907632280261, 9.009237170440699, 9.367922660183149, 9.688276285021297, 10.16240598310771, 10.79740643112537, 11.27748668904297, 11.87640764749929, 12.11961506046011, 12.40794804091680, 13.30356727920397
