| L(s) = 1 | − 3-s − 5-s − 3·7-s + 9-s + 4·11-s − 5·13-s + 15-s + 3·21-s − 2·23-s + 25-s − 27-s − 8·29-s − 7·31-s − 4·33-s + 3·35-s − 7·37-s + 5·39-s − 3·43-s − 45-s + 6·47-s + 2·49-s + 2·53-s − 4·55-s + 61-s − 3·63-s + 5·65-s + 13·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 1.20·11-s − 1.38·13-s + 0.258·15-s + 0.654·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s − 1.25·31-s − 0.696·33-s + 0.507·35-s − 1.15·37-s + 0.800·39-s − 0.457·43-s − 0.149·45-s + 0.875·47-s + 2/7·49-s + 0.274·53-s − 0.539·55-s + 0.128·61-s − 0.377·63-s + 0.620·65-s + 1.58·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4034300983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4034300983\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−12.51664990772377, −12.18503783574070, −11.67288922213763, −11.29489049742505, −10.81808243438139, −10.16998633909102, −9.896651979826937, −9.336898020903225, −9.110770318608718, −8.549853812572497, −7.757372398631563, −7.380972972229436, −6.953308355974931, −6.618802436010817, −6.071533323543981, −5.489364631581233, −5.155615000828777, −4.448903787741572, −3.906150540035188, −3.569926113598270, −3.073899271802217, −2.153756538034021, −1.851184215683537, −0.9022736463602779, −0.2045672449358928,
0.2045672449358928, 0.9022736463602779, 1.851184215683537, 2.153756538034021, 3.073899271802217, 3.569926113598270, 3.906150540035188, 4.448903787741572, 5.155615000828777, 5.489364631581233, 6.071533323543981, 6.618802436010817, 6.953308355974931, 7.380972972229436, 7.757372398631563, 8.549853812572497, 9.110770318608718, 9.336898020903225, 9.896651979826937, 10.16998633909102, 10.81808243438139, 11.29489049742505, 11.67288922213763, 12.18503783574070, 12.51664990772377
