| L(s) = 1 | + 3-s − 2·7-s + 9-s − 11-s − 6·13-s + 4·17-s − 2·19-s − 2·21-s + 8·23-s + 27-s − 33-s + 6·37-s − 6·39-s − 10·43-s − 3·49-s + 4·51-s − 14·53-s − 2·57-s − 12·59-s − 14·61-s − 2·63-s − 4·67-s + 8·69-s − 6·73-s + 2·77-s + 2·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.970·17-s − 0.458·19-s − 0.436·21-s + 1.66·23-s + 0.192·27-s − 0.174·33-s + 0.986·37-s − 0.960·39-s − 1.52·43-s − 3/7·49-s + 0.560·51-s − 1.92·53-s − 0.264·57-s − 1.56·59-s − 1.79·61-s − 0.251·63-s − 0.488·67-s + 0.963·69-s − 0.702·73-s + 0.227·77-s + 0.225·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.174461045271428155286511351853, −7.53803840711764935566844456490, −6.92221850855986788238846208999, −6.08197105100066686265971381936, −5.05120723299971057048476955300, −4.47199005078239721584177735997, −3.08608524332757317521539271782, −2.91838551206243985528277189809, −1.57326728327252056716049122705, 0,
1.57326728327252056716049122705, 2.91838551206243985528277189809, 3.08608524332757317521539271782, 4.47199005078239721584177735997, 5.05120723299971057048476955300, 6.08197105100066686265971381936, 6.92221850855986788238846208999, 7.53803840711764935566844456490, 8.174461045271428155286511351853
