| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 2·11-s − 2·14-s + 16-s − 6·19-s + 2·22-s − 3·23-s − 2·28-s + 6·29-s + 6·31-s + 32-s + 5·37-s − 6·38-s + 9·41-s − 6·43-s + 2·44-s − 3·46-s − 2·47-s − 3·49-s + 3·53-s − 2·56-s + 6·58-s + 9·59-s − 7·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.603·11-s − 0.534·14-s + 1/4·16-s − 1.37·19-s + 0.426·22-s − 0.625·23-s − 0.377·28-s + 1.11·29-s + 1.07·31-s + 0.176·32-s + 0.821·37-s − 0.973·38-s + 1.40·41-s − 0.914·43-s + 0.301·44-s − 0.442·46-s − 0.291·47-s − 3/7·49-s + 0.412·53-s − 0.267·56-s + 0.787·58-s + 1.17·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.69561438651661, −13.19174108679523, −12.83975114862769, −12.34606679752292, −11.80861079849770, −11.59100274753904, −10.76209843333976, −10.42617948090918, −9.964460253220834, −9.355689777070153, −8.924699012234950, −8.171107971205463, −7.951563318839657, −7.090252464328071, −6.600052417457421, −6.253136120871961, −5.921697041528885, −5.106618769306822, −4.452136610579136, −4.175239968629694, −3.557866145446313, −2.832361883299791, −2.483589018666930, −1.667898298577334, −0.9225205285923764, 0,
0.9225205285923764, 1.667898298577334, 2.483589018666930, 2.832361883299791, 3.557866145446313, 4.175239968629694, 4.452136610579136, 5.106618769306822, 5.921697041528885, 6.253136120871961, 6.600052417457421, 7.090252464328071, 7.951563318839657, 8.171107971205463, 8.924699012234950, 9.355689777070153, 9.964460253220834, 10.42617948090918, 10.76209843333976, 11.59100274753904, 11.80861079849770, 12.34606679752292, 12.83975114862769, 13.19174108679523, 13.69561438651661
