| L(s) = 1 | + 3-s − 3·5-s − 2·9-s − 11-s + 6·13-s − 3·15-s − 6·19-s − 3·23-s + 4·25-s − 5·27-s + 3·31-s − 33-s + 37-s + 6·39-s + 10·41-s + 2·43-s + 6·45-s − 12·47-s + 6·53-s + 3·55-s − 6·57-s + 11·59-s + 4·61-s − 18·65-s − 7·67-s − 3·69-s − 13·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 2/3·9-s − 0.301·11-s + 1.66·13-s − 0.774·15-s − 1.37·19-s − 0.625·23-s + 4/5·25-s − 0.962·27-s + 0.538·31-s − 0.174·33-s + 0.164·37-s + 0.960·39-s + 1.56·41-s + 0.304·43-s + 0.894·45-s − 1.75·47-s + 0.824·53-s + 0.404·55-s − 0.794·57-s + 1.43·59-s + 0.512·61-s − 2.23·65-s − 0.855·67-s − 0.361·69-s − 1.54·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 17 T + p T^{2} \) | 1.89.ar |
| 97 | \( 1 - 15 T + p T^{2} \) | 1.97.ap |
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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
−15.25627135898628, −14.69478397566329, −14.32089030750765, −13.67009315778345, −12.96954359131454, −12.88201345975397, −11.75813823643846, −11.68409900137520, −11.02893380667561, −10.60517142528494, −9.943389631502168, −9.012312063169212, −8.710659378702441, −8.133370868458757, −7.940439428327295, −7.206661083049951, −6.333071158183261, −6.057967195069672, −5.205068726757196, −4.277698732403842, −3.963140408238883, −3.397000934614680, −2.698477485225984, −1.961583805015527, −0.8779389424760439, 0,
0.8779389424760439, 1.961583805015527, 2.698477485225984, 3.397000934614680, 3.963140408238883, 4.277698732403842, 5.205068726757196, 6.057967195069672, 6.333071158183261, 7.206661083049951, 7.940439428327295, 8.133370868458757, 8.710659378702441, 9.012312063169212, 9.943389631502168, 10.60517142528494, 11.02893380667561, 11.68409900137520, 11.75813823643846, 12.88201345975397, 12.96954359131454, 13.67009315778345, 14.32089030750765, 14.69478397566329, 15.25627135898628
