| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 4·11-s − 6·13-s + 15-s + 6·17-s + 8·19-s + 21-s + 8·23-s + 25-s + 27-s − 2·29-s + 4·33-s + 35-s + 2·37-s − 6·39-s − 2·41-s − 8·43-s + 45-s + 12·47-s + 49-s + 6·51-s + 53-s + 4·55-s + 8·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.258·15-s + 1.45·17-s + 1.83·19-s + 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.696·33-s + 0.169·35-s + 0.328·37-s − 0.960·39-s − 0.312·41-s − 1.21·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.137·53-s + 0.539·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.871387014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.871387014\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 53 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.57707193732481, −14.23655888441262, −13.79896157031052, −13.19286055152792, −12.52124249695692, −12.09033320828886, −11.65267456577895, −11.13914934106461, −10.17609876414191, −9.904097338381138, −9.437603042524874, −8.983648952092934, −8.362808295880746, −7.580506345279594, −7.187474638510498, −6.933374507632336, −5.849360569769320, −5.357700189829622, −4.899363237408560, −4.177109063003918, −3.310734284897200, −2.994606317012510, −2.170498500831170, −1.338595670813115, −0.8572199997483300,
0.8572199997483300, 1.338595670813115, 2.170498500831170, 2.994606317012510, 3.310734284897200, 4.177109063003918, 4.899363237408560, 5.357700189829622, 5.849360569769320, 6.933374507632336, 7.187474638510498, 7.580506345279594, 8.362808295880746, 8.983648952092934, 9.437603042524874, 9.904097338381138, 10.17609876414191, 11.13914934106461, 11.65267456577895, 12.09033320828886, 12.52124249695692, 13.19286055152792, 13.79896157031052, 14.23655888441262, 14.57707193732481
