| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 4·13-s + 15-s − 6·17-s + 4·19-s + 2·21-s − 4·23-s + 25-s − 27-s + 4·29-s + 4·31-s + 2·35-s − 2·37-s − 4·39-s − 41-s + 4·43-s − 45-s − 6·47-s − 3·49-s + 6·51-s + 4·53-s − 4·57-s + 4·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.10·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 0.718·31-s + 0.338·35-s − 0.328·37-s − 0.640·39-s − 0.156·41-s + 0.609·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.840·51-s + 0.549·53-s − 0.529·57-s + 0.520·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 41 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.41724616552316, −14.51607506719435, −13.83285436127454, −13.52059452481742, −12.99817808494349, −12.38140484005035, −11.94689914305397, −11.31955634746540, −11.04823176270648, −10.29502243919027, −9.936718643850909, −9.175293205720376, −8.744260948289270, −8.093181887813400, −7.536958996685301, −6.709569669101204, −6.465023908783225, −5.930579891895454, −5.161570212993221, −4.517663469205175, −3.945523829833106, −3.335240521120769, −2.636779362175478, −1.695528877199412, −0.8346758518810117, 0,
0.8346758518810117, 1.695528877199412, 2.636779362175478, 3.335240521120769, 3.945523829833106, 4.517663469205175, 5.161570212993221, 5.930579891895454, 6.465023908783225, 6.709569669101204, 7.536958996685301, 8.093181887813400, 8.744260948289270, 9.175293205720376, 9.936718643850909, 10.29502243919027, 11.04823176270648, 11.31955634746540, 11.94689914305397, 12.38140484005035, 12.99817808494349, 13.52059452481742, 13.83285436127454, 14.51607506719435, 15.41724616552316
