| L(s) = 1 | + 3-s + 2·7-s + 9-s + 4·11-s − 13-s − 8·17-s − 6·19-s + 2·21-s + 6·23-s + 27-s + 4·29-s + 4·33-s − 2·37-s − 39-s − 2·41-s + 4·43-s − 3·49-s − 8·51-s − 10·53-s − 6·57-s + 4·59-s + 10·61-s + 2·63-s − 12·67-s + 6·69-s + 8·71-s + 8·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 1.94·17-s − 1.37·19-s + 0.436·21-s + 1.25·23-s + 0.192·27-s + 0.742·29-s + 0.696·33-s − 0.328·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s − 3/7·49-s − 1.12·51-s − 1.37·53-s − 0.794·57-s + 0.520·59-s + 1.28·61-s + 0.251·63-s − 1.46·67-s + 0.722·69-s + 0.949·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 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 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−14.45936768846967, −14.22731550044571, −13.50740343900434, −12.99474220454888, −12.69552943222671, −11.90954673979685, −11.48050602363875, −10.89880026934887, −10.66100524789105, −9.820220766064453, −9.185972985549463, −8.922697619912190, −8.360694977376551, −8.022895677981988, −6.979911033406157, −6.840636842040231, −6.342620700373621, −5.471264962961606, −4.624233397766230, −4.463882144199261, −3.856086483714176, −3.011987318142309, −2.345949590442893, −1.797389169948226, −1.115452968796426, 0,
1.115452968796426, 1.797389169948226, 2.345949590442893, 3.011987318142309, 3.856086483714176, 4.463882144199261, 4.624233397766230, 5.471264962961606, 6.342620700373621, 6.840636842040231, 6.979911033406157, 8.022895677981988, 8.360694977376551, 8.922697619912190, 9.185972985549463, 9.820220766064453, 10.66100524789105, 10.89880026934887, 11.48050602363875, 11.90954673979685, 12.69552943222671, 12.99474220454888, 13.50740343900434, 14.22731550044571, 14.45936768846967
