| L(s) = 1 | − 3-s − 3·7-s + 9-s − 11-s + 4·13-s − 17-s + 2·19-s + 3·21-s + 6·23-s − 5·25-s − 27-s + 9·29-s − 4·31-s + 33-s − 2·37-s − 4·39-s − 41-s − 6·43-s − 9·47-s + 2·49-s + 51-s − 13·53-s − 2·57-s + 15·59-s + 14·61-s − 3·63-s + 9·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.242·17-s + 0.458·19-s + 0.654·21-s + 1.25·23-s − 25-s − 0.192·27-s + 1.67·29-s − 0.718·31-s + 0.174·33-s − 0.328·37-s − 0.640·39-s − 0.156·41-s − 0.914·43-s − 1.31·47-s + 2/7·49-s + 0.140·51-s − 1.78·53-s − 0.264·57-s + 1.95·59-s + 1.79·61-s − 0.377·63-s + 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 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
−15.41335577176056, −14.62234336617714, −14.12477471568447, −13.31347769625448, −13.07532995499012, −12.80428881909773, −11.84298594457399, −11.58169530249083, −11.00344316318834, −10.33909787844467, −9.941471204735475, −9.453411654579072, −8.661640248855180, −8.348924407902968, −7.449797545617749, −6.884753049144314, −6.387916015038029, −5.991413926941950, −5.189244600151347, −4.770386959019987, −3.779708537897510, −3.384396146577139, −2.707976694135815, −1.696405196730160, −0.8889191437703834, 0,
0.8889191437703834, 1.696405196730160, 2.707976694135815, 3.384396146577139, 3.779708537897510, 4.770386959019987, 5.189244600151347, 5.991413926941950, 6.387916015038029, 6.884753049144314, 7.449797545617749, 8.348924407902968, 8.661640248855180, 9.453411654579072, 9.941471204735475, 10.33909787844467, 11.00344316318834, 11.58169530249083, 11.84298594457399, 12.80428881909773, 13.07532995499012, 13.31347769625448, 14.12477471568447, 14.62234336617714, 15.41335577176056
