| L(s) = 1 | − 3-s − 2·7-s − 2·9-s − 3·17-s − 8·19-s + 2·21-s − 3·23-s − 5·25-s + 5·27-s + 9·29-s + 8·31-s − 8·37-s − 6·41-s − 43-s + 12·47-s − 3·49-s + 3·51-s − 3·53-s + 8·57-s + 6·59-s + 7·61-s + 4·63-s + 14·67-s + 3·69-s − 12·71-s + 2·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s − 0.727·17-s − 1.83·19-s + 0.436·21-s − 0.625·23-s − 25-s + 0.962·27-s + 1.67·29-s + 1.43·31-s − 1.31·37-s − 0.937·41-s − 0.152·43-s + 1.75·47-s − 3/7·49-s + 0.420·51-s − 0.412·53-s + 1.05·57-s + 0.781·59-s + 0.896·61-s + 0.503·63-s + 1.71·67-s + 0.361·69-s − 1.42·71-s + 0.234·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−12.67405878261075, −12.32330122108296, −11.99466438173860, −11.49794959870100, −11.03830408854703, −10.52798378777853, −10.08464659612030, −9.923324138404246, −9.078464217904882, −8.610105396672218, −8.352202678722776, −7.951344132452138, −6.875324031941692, −6.804283860387433, −6.331446639249190, −5.890062364804135, −5.413825260738984, −4.735740011103833, −4.335689716760119, −3.804774977433367, −3.194144860054319, −2.455647515369849, −2.284237780901635, −1.358102429906755, −0.5005988691734064, 0,
0.5005988691734064, 1.358102429906755, 2.284237780901635, 2.455647515369849, 3.194144860054319, 3.804774977433367, 4.335689716760119, 4.735740011103833, 5.413825260738984, 5.890062364804135, 6.331446639249190, 6.804283860387433, 6.875324031941692, 7.951344132452138, 8.352202678722776, 8.610105396672218, 9.078464217904882, 9.923324138404246, 10.08464659612030, 10.52798378777853, 11.03830408854703, 11.49794959870100, 11.99466438173860, 12.32330122108296, 12.67405878261075
