| L(s) = 1 | + 7-s − 3·9-s − 2·13-s − 17-s + 4·19-s + 2·29-s − 8·31-s − 2·37-s − 2·41-s − 4·43-s − 4·47-s + 49-s + 2·53-s + 12·59-s + 10·61-s − 3·63-s − 4·67-s + 16·71-s − 2·73-s − 16·79-s + 9·81-s + 12·83-s + 2·89-s − 2·91-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s − 0.554·13-s − 0.242·17-s + 0.917·19-s + 0.371·29-s − 1.43·31-s − 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.274·53-s + 1.56·59-s + 1.28·61-s − 0.377·63-s − 0.488·67-s + 1.89·71-s − 0.234·73-s − 1.80·79-s + 81-s + 1.31·83-s + 0.211·89-s − 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.353154597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.353154597\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.28774379598957, −12.56491510188651, −12.05827771265283, −11.75130446192485, −11.16543473705776, −10.97803513597784, −10.20481786818171, −9.851855613087802, −9.249817964460827, −8.874968682455309, −8.230576214492928, −8.023972761590611, −7.286172134511400, −6.886282067833340, −6.398140211344272, −5.559990445703768, −5.336165123684918, −4.963433033351633, −4.111744475144346, −3.644482356791506, −3.029638503047547, −2.469766163757429, −1.914052186540987, −1.164170863036927, −0.3417998806033155,
0.3417998806033155, 1.164170863036927, 1.914052186540987, 2.469766163757429, 3.029638503047547, 3.644482356791506, 4.111744475144346, 4.963433033351633, 5.336165123684918, 5.559990445703768, 6.398140211344272, 6.886282067833340, 7.286172134511400, 8.023972761590611, 8.230576214492928, 8.874968682455309, 9.249817964460827, 9.851855613087802, 10.20481786818171, 10.97803513597784, 11.16543473705776, 11.75130446192485, 12.05827771265283, 12.56491510188651, 13.28774379598957
