| L(s) = 1 | + 3-s − 5-s + 9-s + 2·11-s − 15-s + 2·17-s + 4·19-s + 25-s + 27-s + 10·29-s + 4·31-s + 2·33-s + 8·37-s − 10·41-s − 45-s − 6·47-s − 7·49-s + 2·51-s − 2·53-s − 2·55-s + 4·57-s + 14·59-s + 2·61-s + 8·67-s − 6·71-s + 12·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s − 0.258·15-s + 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.192·27-s + 1.85·29-s + 0.718·31-s + 0.348·33-s + 1.31·37-s − 1.56·41-s − 0.149·45-s − 0.875·47-s − 49-s + 0.280·51-s − 0.274·53-s − 0.269·55-s + 0.529·57-s + 1.82·59-s + 0.256·61-s + 0.977·67-s − 0.712·71-s + 1.40·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.416508942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.416508942\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−14.68290395086958, −14.17779999572036, −13.98457739051676, −13.12557909592426, −12.83444967184233, −12.05814554788285, −11.63045500586025, −11.36247280648461, −10.38724316154250, −9.916827813631106, −9.633061651162674, −8.793195318756286, −8.305304095633452, −8.000912726945523, −7.227039477462263, −6.730164360653624, −6.221533745884908, −5.361555361322364, −4.732857041491377, −4.233283080710694, −3.374903563985706, −3.099632252973519, −2.251035665400183, −1.325265994379525, −0.7153932972498056,
0.7153932972498056, 1.325265994379525, 2.251035665400183, 3.099632252973519, 3.374903563985706, 4.233283080710694, 4.732857041491377, 5.361555361322364, 6.221533745884908, 6.730164360653624, 7.227039477462263, 8.000912726945523, 8.305304095633452, 8.793195318756286, 9.633061651162674, 9.916827813631106, 10.38724316154250, 11.36247280648461, 11.63045500586025, 12.05814554788285, 12.83444967184233, 13.12557909592426, 13.98457739051676, 14.17779999572036, 14.68290395086958
