| L(s) = 1 | − 2-s + 4-s − 8-s + 11-s + 2·13-s + 16-s + 2·17-s + 2·19-s − 22-s + 4·23-s − 2·26-s + 4·29-s − 31-s − 32-s − 2·34-s − 2·38-s − 2·41-s + 4·43-s + 44-s − 4·46-s − 6·47-s − 7·49-s + 2·52-s + 4·53-s − 4·58-s − 12·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.301·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.458·19-s − 0.213·22-s + 0.834·23-s − 0.392·26-s + 0.742·29-s − 0.179·31-s − 0.176·32-s − 0.342·34-s − 0.324·38-s − 0.312·41-s + 0.609·43-s + 0.150·44-s − 0.589·46-s − 0.875·47-s − 49-s + 0.277·52-s + 0.549·53-s − 0.525·58-s − 1.56·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 31 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.69402350822519, −13.00203582121926, −12.50392209524461, −12.09458418027559, −11.57835813701057, −11.06224260462624, −10.74809467771157, −10.16794120958737, −9.638688089201184, −9.280218596383879, −8.739929527429814, −8.301432346188592, −7.746388592420921, −7.344704635441925, −6.706550451933664, −6.294553173758763, −5.796270803357163, −5.091596709921163, −4.661287194409803, −3.875296500148205, −3.251195636450023, −2.907027281922157, −2.051479854653246, −1.368041296230341, −0.9317478095544872, 0,
0.9317478095544872, 1.368041296230341, 2.051479854653246, 2.907027281922157, 3.251195636450023, 3.875296500148205, 4.661287194409803, 5.091596709921163, 5.796270803357163, 6.294553173758763, 6.706550451933664, 7.344704635441925, 7.746388592420921, 8.301432346188592, 8.739929527429814, 9.280218596383879, 9.638688089201184, 10.16794120958737, 10.74809467771157, 11.06224260462624, 11.57835813701057, 12.09458418027559, 12.50392209524461, 13.00203582121926, 13.69402350822519
