| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s − 5·11-s + 4·14-s + 16-s − 2·17-s + 8·19-s + 5·22-s + 3·23-s − 4·28-s − 7·29-s − 7·31-s − 32-s + 2·34-s + 3·37-s − 8·38-s + 43-s − 5·44-s − 3·46-s − 7·47-s + 9·49-s + 10·53-s + 4·56-s + 7·58-s + 5·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 1.50·11-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 1.83·19-s + 1.06·22-s + 0.625·23-s − 0.755·28-s − 1.29·29-s − 1.25·31-s − 0.176·32-s + 0.342·34-s + 0.493·37-s − 1.29·38-s + 0.152·43-s − 0.753·44-s − 0.442·46-s − 1.02·47-s + 9/7·49-s + 1.37·53-s + 0.534·56-s + 0.919·58-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.45672418688319, −13.42829547424389, −13.25592749169756, −12.99873674549831, −12.31264304409632, −11.73049493048598, −11.20080711509513, −10.71244108299135, −10.15189739565555, −9.759261660672540, −9.286104709253929, −8.926630249043574, −8.151991832449586, −7.603257444154652, −7.129593930402993, −6.852005831101635, −5.920034253675211, −5.562260583391394, −5.118614562659767, −4.116259500815951, −3.405538799359390, −2.974399745972141, −2.468062111172576, −1.618103985701596, −0.6416535740881581, 0,
0.6416535740881581, 1.618103985701596, 2.468062111172576, 2.974399745972141, 3.405538799359390, 4.116259500815951, 5.118614562659767, 5.562260583391394, 5.920034253675211, 6.852005831101635, 7.129593930402993, 7.603257444154652, 8.151991832449586, 8.926630249043574, 9.286104709253929, 9.759261660672540, 10.15189739565555, 10.71244108299135, 11.20080711509513, 11.73049493048598, 12.31264304409632, 12.99873674549831, 13.25592749169756, 13.42829547424389, 14.45672418688319
