| L(s) = 1 | + 2·7-s − 11-s + 4·13-s + 2·17-s − 2·23-s − 5·25-s + 2·29-s + 4·31-s + 6·37-s + 6·41-s + 12·43-s − 6·47-s − 3·49-s − 4·67-s − 10·71-s + 2·73-s − 2·77-s − 2·79-s − 4·83-s + 14·89-s + 8·91-s − 2·97-s + 10·101-s − 8·103-s + 4·107-s + 20·109-s + 14·113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s + 1.10·13-s + 0.485·17-s − 0.417·23-s − 25-s + 0.371·29-s + 0.718·31-s + 0.986·37-s + 0.937·41-s + 1.82·43-s − 0.875·47-s − 3/7·49-s − 0.488·67-s − 1.18·71-s + 0.234·73-s − 0.227·77-s − 0.225·79-s − 0.439·83-s + 1.48·89-s + 0.838·91-s − 0.203·97-s + 0.995·101-s − 0.788·103-s + 0.386·107-s + 1.91·109-s + 1.31·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.162950363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.162950363\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.539229320199990255953113353535, −7.946924210891244364126307750805, −7.39938134101451624136393934258, −6.17158997235851069613217629810, −5.82427774140814655002360334367, −4.72985157824507529825380135842, −4.06987624784385777482324026631, −3.06286335678909796832639839808, −1.98827265692693803354483679246, −0.929541519182740551444742608914,
0.929541519182740551444742608914, 1.98827265692693803354483679246, 3.06286335678909796832639839808, 4.06987624784385777482324026631, 4.72985157824507529825380135842, 5.82427774140814655002360334367, 6.17158997235851069613217629810, 7.39938134101451624136393934258, 7.946924210891244364126307750805, 8.539229320199990255953113353535
