| L(s) = 1 | + 2-s − 4-s − 3·8-s − 16-s − 2·17-s − 4·19-s + 6·23-s − 5·25-s − 4·29-s + 5·32-s − 2·34-s + 10·37-s − 4·38-s + 12·41-s − 4·43-s + 6·46-s − 10·47-s − 5·50-s − 12·53-s − 4·58-s − 14·59-s + 10·61-s + 7·64-s + 2·68-s − 8·71-s + 2·73-s + 10·74-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1/4·16-s − 0.485·17-s − 0.917·19-s + 1.25·23-s − 25-s − 0.742·29-s + 0.883·32-s − 0.342·34-s + 1.64·37-s − 0.648·38-s + 1.87·41-s − 0.609·43-s + 0.884·46-s − 1.45·47-s − 0.707·50-s − 1.64·53-s − 0.525·58-s − 1.82·59-s + 1.28·61-s + 7/8·64-s + 0.242·68-s − 0.949·71-s + 0.234·73-s + 1.16·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.381344571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.381344571\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−14.13688170050436, −13.40038810308855, −13.07708682916360, −12.79597791531004, −12.30017383940589, −11.43944203093884, −11.28517678842542, −10.72261737625521, −9.859221714468182, −9.551224551253492, −9.025032993662848, −8.563414431503675, −7.855587739551667, −7.515647671203320, −6.542930542436213, −6.270173982194655, −5.699218003153217, −5.051705169649308, −4.477246880069562, −4.178047022439526, −3.409959617464624, −2.873648031395085, −2.188831817853934, −1.329583961835285, −0.3513626754533672,
0.3513626754533672, 1.329583961835285, 2.188831817853934, 2.873648031395085, 3.409959617464624, 4.178047022439526, 4.477246880069562, 5.051705169649308, 5.699218003153217, 6.270173982194655, 6.542930542436213, 7.515647671203320, 7.855587739551667, 8.563414431503675, 9.025032993662848, 9.551224551253492, 9.859221714468182, 10.72261737625521, 11.28517678842542, 11.43944203093884, 12.30017383940589, 12.79597791531004, 13.07708682916360, 13.40038810308855, 14.13688170050436
