| L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s + 2·13-s − 2·15-s − 21-s − 25-s − 27-s − 2·29-s − 8·31-s + 2·35-s + 6·37-s − 2·39-s + 6·41-s + 4·43-s + 2·45-s + 49-s + 14·53-s − 8·59-s − 14·61-s + 63-s + 4·65-s + 4·67-s − 8·71-s − 10·73-s + 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.338·35-s + 0.986·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 1.92·53-s − 1.04·59-s − 1.79·61-s + 0.125·63-s + 0.496·65-s + 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 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
−14.76286500458865, −14.24988315980789, −13.79815408600460, −13.16715688251758, −12.88712563485588, −12.28947115939984, −11.53455582062971, −11.30905000757613, −10.60835261662634, −10.28148587020334, −9.633057790320234, −9.012646577941604, −8.785759417078545, −7.677817805241622, −7.548219540043329, −6.734925162346707, −6.005277466498207, −5.784536242270535, −5.279320348813732, −4.396255433607748, −4.049072849411868, −3.117393394104462, −2.363342292608168, −1.670978730684775, −1.092083191169642, 0,
1.092083191169642, 1.670978730684775, 2.363342292608168, 3.117393394104462, 4.049072849411868, 4.396255433607748, 5.279320348813732, 5.784536242270535, 6.005277466498207, 6.734925162346707, 7.548219540043329, 7.677817805241622, 8.785759417078545, 9.012646577941604, 9.633057790320234, 10.28148587020334, 10.60835261662634, 11.30905000757613, 11.53455582062971, 12.28947115939984, 12.88712563485588, 13.16715688251758, 13.79815408600460, 14.24988315980789, 14.76286500458865
