| L(s) = 1 | − 7-s − 4·11-s − 3·13-s + 4·17-s + 19-s + 8·29-s − 31-s + 2·37-s − 2·41-s + 11·43-s − 2·47-s − 6·49-s + 10·53-s − 6·59-s + 11·61-s − 9·67-s + 6·71-s − 14·73-s + 4·77-s − 16·79-s + 2·83-s + 3·91-s − 11·97-s − 14·101-s − 8·103-s + 6·107-s − 11·109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s − 0.832·13-s + 0.970·17-s + 0.229·19-s + 1.48·29-s − 0.179·31-s + 0.328·37-s − 0.312·41-s + 1.67·43-s − 0.291·47-s − 6/7·49-s + 1.37·53-s − 0.781·59-s + 1.40·61-s − 1.09·67-s + 0.712·71-s − 1.63·73-s + 0.455·77-s − 1.80·79-s + 0.219·83-s + 0.314·91-s − 1.11·97-s − 1.39·101-s − 0.788·103-s + 0.580·107-s − 1.05·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−7.55990952528509660184379426977, −7.02483085598777485384329535953, −6.09711061573529741618994459468, −5.44002752439634942979028637537, −4.83122606304056188900417487383, −3.97077702766610761045086504504, −2.90773431001244068562128203027, −2.55302736290516311687659169676, −1.19868955287852667378705259965, 0,
1.19868955287852667378705259965, 2.55302736290516311687659169676, 2.90773431001244068562128203027, 3.97077702766610761045086504504, 4.83122606304056188900417487383, 5.44002752439634942979028637537, 6.09711061573529741618994459468, 7.02483085598777485384329535953, 7.55990952528509660184379426977
