| L(s) = 1 | − 2·5-s − 7-s − 13-s + 6·17-s − 4·19-s − 25-s + 2·29-s + 2·35-s − 2·37-s − 10·41-s + 8·43-s + 49-s + 2·53-s − 12·59-s − 6·61-s + 2·65-s − 8·67-s + 8·71-s − 14·73-s + 8·79-s + 4·83-s − 12·85-s + 14·89-s + 91-s + 8·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 0.277·13-s + 1.45·17-s − 0.917·19-s − 1/5·25-s + 0.371·29-s + 0.338·35-s − 0.328·37-s − 1.56·41-s + 1.21·43-s + 1/7·49-s + 0.274·53-s − 1.56·59-s − 0.768·61-s + 0.248·65-s − 0.977·67-s + 0.949·71-s − 1.63·73-s + 0.900·79-s + 0.439·83-s − 1.30·85-s + 1.48·89-s + 0.104·91-s + 0.820·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.065601518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.065601518\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−15.32177885372778, −14.84433256598131, −14.31988064100178, −13.71715715036763, −13.14520336733673, −12.47242122862574, −11.99716317378391, −11.84014831580464, −10.84031169182297, −10.52538797081799, −9.895751333213720, −9.289442582941196, −8.652588358385395, −8.055230741308396, −7.565046176150128, −7.086960194247743, −6.252223972820705, −5.825792549781138, −4.935239122682602, −4.434457239198220, −3.596268185151842, −3.261765301493277, −2.354954981862557, −1.427106973141308, −0.4156684821831894,
0.4156684821831894, 1.427106973141308, 2.354954981862557, 3.261765301493277, 3.596268185151842, 4.434457239198220, 4.935239122682602, 5.825792549781138, 6.252223972820705, 7.086960194247743, 7.565046176150128, 8.055230741308396, 8.652588358385395, 9.289442582941196, 9.895751333213720, 10.52538797081799, 10.84031169182297, 11.84014831580464, 11.99716317378391, 12.47242122862574, 13.14520336733673, 13.71715715036763, 14.31988064100178, 14.84433256598131, 15.32177885372778
