| L(s) = 1 | + 3-s − 3·5-s + 7-s + 9-s + 11-s − 3·13-s − 3·15-s + 4·17-s − 7·19-s + 21-s − 2·23-s + 4·25-s + 27-s − 3·29-s + 33-s − 3·35-s − 11·37-s − 3·39-s + 8·41-s − 2·43-s − 3·45-s − 9·47-s + 49-s + 4·51-s + 10·53-s − 3·55-s − 7·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.832·13-s − 0.774·15-s + 0.970·17-s − 1.60·19-s + 0.218·21-s − 0.417·23-s + 4/5·25-s + 0.192·27-s − 0.557·29-s + 0.174·33-s − 0.507·35-s − 1.80·37-s − 0.480·39-s + 1.24·41-s − 0.304·43-s − 0.447·45-s − 1.31·47-s + 1/7·49-s + 0.560·51-s + 1.37·53-s − 0.404·55-s − 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−8.635530002479711911807663353444, −8.082864567579570827623668611244, −7.45651225258303462325362670852, −6.74130034170550166259021910916, −5.51633184042291595562202569794, −4.44511078048048553410871379288, −3.90719150030222637297333614085, −2.93531963849626206594334394511, −1.71748086423211192776019423595, 0,
1.71748086423211192776019423595, 2.93531963849626206594334394511, 3.90719150030222637297333614085, 4.44511078048048553410871379288, 5.51633184042291595562202569794, 6.74130034170550166259021910916, 7.45651225258303462325362670852, 8.082864567579570827623668611244, 8.635530002479711911807663353444
