L(s) = 1 | + 2-s + 4-s − 3·7-s + 8-s − 4·13-s − 3·14-s + 16-s + 3·17-s + 5·19-s − 4·23-s − 4·26-s − 3·28-s + 5·29-s + 7·31-s + 32-s + 3·34-s − 7·37-s + 5·38-s − 8·41-s + 6·43-s − 4·46-s − 8·47-s + 2·49-s − 4·52-s − 9·53-s − 3·56-s + 5·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s − 1.10·13-s − 0.801·14-s + 1/4·16-s + 0.727·17-s + 1.14·19-s − 0.834·23-s − 0.784·26-s − 0.566·28-s + 0.928·29-s + 1.25·31-s + 0.176·32-s + 0.514·34-s − 1.15·37-s + 0.811·38-s − 1.24·41-s + 0.914·43-s − 0.589·46-s − 1.16·47-s + 2/7·49-s − 0.554·52-s − 1.23·53-s − 0.400·56-s + 0.656·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.48769654780839, −14.18635660053701, −13.63624853186846, −13.25702036704339, −12.49880395237466, −12.21130253749514, −11.88533135606902, −11.28365076675679, −10.40420887067824, −9.979281270731358, −9.796757107310993, −9.101478804614933, −8.273993421393287, −7.800826229105563, −7.208528099469020, −6.598230698270335, −6.292609782757892, −5.495308069228840, −5.042338763367052, −4.494563185722894, −3.615874631480411, −3.209854864730566, −2.701229968275976, −1.905786827475247, −0.9683303596213926, 0,
0.9683303596213926, 1.905786827475247, 2.701229968275976, 3.209854864730566, 3.615874631480411, 4.494563185722894, 5.042338763367052, 5.495308069228840, 6.292609782757892, 6.598230698270335, 7.208528099469020, 7.800826229105563, 8.273993421393287, 9.101478804614933, 9.796757107310993, 9.979281270731358, 10.40420887067824, 11.28365076675679, 11.88533135606902, 12.21130253749514, 12.49880395237466, 13.25702036704339, 13.63624853186846, 14.18635660053701, 14.48769654780839