| L(s) = 1 | − 3-s + 9-s + 11-s − 3·17-s − 3·19-s − 3·23-s − 5·25-s − 27-s + 3·29-s + 6·31-s − 33-s + 5·37-s − 2·41-s + 7·43-s + 3·47-s + 3·51-s − 12·53-s + 3·57-s − 5·59-s + 2·61-s + 4·67-s + 3·69-s − 13·71-s + 8·73-s + 5·75-s − 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.727·17-s − 0.688·19-s − 0.625·23-s − 25-s − 0.192·27-s + 0.557·29-s + 1.07·31-s − 0.174·33-s + 0.821·37-s − 0.312·41-s + 1.06·43-s + 0.437·47-s + 0.420·51-s − 1.64·53-s + 0.397·57-s − 0.650·59-s + 0.256·61-s + 0.488·67-s + 0.361·69-s − 1.54·71-s + 0.936·73-s + 0.577·75-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−14.71757376043026, −14.13171022837877, −13.80094794373224, −13.07214381101228, −12.74733358820952, −12.06890910530216, −11.70972359011369, −11.15725821615028, −10.71066324098931, −10.03091611583777, −9.742427883536394, −8.956030656278692, −8.553783541483416, −7.779269106703190, −7.463282316654316, −6.505580532714790, −6.304149553420006, −5.800636769693996, −4.951430382542659, −4.405275802944769, −4.036135872158302, −3.151976006354471, −2.374785237967351, −1.756395997456913, −0.8559121604235135, 0,
0.8559121604235135, 1.756395997456913, 2.374785237967351, 3.151976006354471, 4.036135872158302, 4.405275802944769, 4.951430382542659, 5.800636769693996, 6.304149553420006, 6.505580532714790, 7.463282316654316, 7.779269106703190, 8.553783541483416, 8.956030656278692, 9.742427883536394, 10.03091611583777, 10.71066324098931, 11.15725821615028, 11.70972359011369, 12.06890910530216, 12.74733358820952, 13.07214381101228, 13.80094794373224, 14.13171022837877, 14.71757376043026
