| L(s) = 1 | − 4·5-s + 2·7-s + 2·13-s + 4·17-s − 6·19-s − 23-s + 11·25-s − 10·29-s + 4·31-s − 8·35-s − 2·37-s + 6·41-s − 6·43-s − 8·47-s − 3·49-s − 8·53-s + 4·59-s − 2·61-s − 8·65-s + 6·67-s − 14·73-s − 10·79-s − 16·83-s − 16·85-s + 4·91-s + 24·95-s − 18·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.755·7-s + 0.554·13-s + 0.970·17-s − 1.37·19-s − 0.208·23-s + 11/5·25-s − 1.85·29-s + 0.718·31-s − 1.35·35-s − 0.328·37-s + 0.937·41-s − 0.914·43-s − 1.16·47-s − 3/7·49-s − 1.09·53-s + 0.520·59-s − 0.256·61-s − 0.992·65-s + 0.733·67-s − 1.63·73-s − 1.12·79-s − 1.75·83-s − 1.73·85-s + 0.419·91-s + 2.46·95-s − 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−8.605756147782062852903604944448, −8.180112487664392685124009980414, −7.58531333009871179983901952118, −6.74232627265011018133251955085, −5.63545927802732522211214526310, −4.57387757240098741482768846387, −3.97313545447119724470253775739, −3.11266023045401698411044468700, −1.54331063307051824363877850466, 0,
1.54331063307051824363877850466, 3.11266023045401698411044468700, 3.97313545447119724470253775739, 4.57387757240098741482768846387, 5.63545927802732522211214526310, 6.74232627265011018133251955085, 7.58531333009871179983901952118, 8.180112487664392685124009980414, 8.605756147782062852903604944448
