| L(s) = 1 | − 6·11-s − 2·13-s − 4·17-s − 2·19-s − 8·23-s − 6·29-s − 8·37-s + 2·41-s + 43-s + 8·47-s − 7·49-s + 4·53-s − 10·59-s + 10·61-s + 4·67-s + 4·71-s + 8·79-s + 8·83-s − 2·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.80·11-s − 0.554·13-s − 0.970·17-s − 0.458·19-s − 1.66·23-s − 1.11·29-s − 1.31·37-s + 0.312·41-s + 0.152·43-s + 1.16·47-s − 49-s + 0.549·53-s − 1.30·59-s + 1.28·61-s + 0.488·67-s + 0.474·71-s + 0.900·79-s + 0.878·83-s − 0.211·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.54614438917720, −13.01250580238463, −12.63292063829236, −12.28575519115616, −11.53545428599116, −11.21666938274926, −10.52397680086096, −10.30820064404343, −9.876374455877090, −9.125745450882376, −8.785839984525135, −8.137364475740038, −7.624853057678322, −7.478796049609229, −6.620025508468380, −6.205935342096519, −5.545215295330749, −5.132271006350443, −4.651360051199534, −3.961206452893188, −3.486432311186937, −2.650666035885076, −2.160279036308988, −1.871317074577057, −0.5538202913269614, 0,
0.5538202913269614, 1.871317074577057, 2.160279036308988, 2.650666035885076, 3.486432311186937, 3.961206452893188, 4.651360051199534, 5.132271006350443, 5.545215295330749, 6.205935342096519, 6.620025508468380, 7.478796049609229, 7.624853057678322, 8.137364475740038, 8.785839984525135, 9.125745450882376, 9.876374455877090, 10.30820064404343, 10.52397680086096, 11.21666938274926, 11.53545428599116, 12.28575519115616, 12.63292063829236, 13.01250580238463, 13.54614438917720
