| L(s) = 1 | − 2-s + 4-s + 2·5-s + 3·7-s − 8-s − 2·10-s + 4·11-s − 4·13-s − 3·14-s + 16-s − 7·17-s − 6·19-s + 2·20-s − 4·22-s − 9·23-s − 25-s + 4·26-s + 3·28-s − 2·29-s − 4·31-s − 32-s + 7·34-s + 6·35-s − 2·37-s + 6·38-s − 2·40-s − 11·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s + 1.13·7-s − 0.353·8-s − 0.632·10-s + 1.20·11-s − 1.10·13-s − 0.801·14-s + 1/4·16-s − 1.69·17-s − 1.37·19-s + 0.447·20-s − 0.852·22-s − 1.87·23-s − 1/5·25-s + 0.784·26-s + 0.566·28-s − 0.371·29-s − 0.718·31-s − 0.176·32-s + 1.20·34-s + 1.01·35-s − 0.328·37-s + 0.973·38-s − 0.316·40-s − 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7112209179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7112209179\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 79 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.82017189390500, −13.23341446484385, −12.45482977843438, −12.09947899801307, −11.71507620643935, −11.00933336273420, −10.73775309544425, −10.21167590334557, −9.630853948648409, −9.164108471683325, −8.800302772004546, −8.337484955452514, −7.628754487387273, −7.294180282214006, −6.525769543485454, −6.189299260257718, −5.748799712624497, −4.876002358437818, −4.403077050047546, −4.025434063580246, −3.023333428228494, −2.085587268145103, −1.913732699864690, −1.602487037706630, −0.2630933467500610,
0.2630933467500610, 1.602487037706630, 1.913732699864690, 2.085587268145103, 3.023333428228494, 4.025434063580246, 4.403077050047546, 4.876002358437818, 5.748799712624497, 6.189299260257718, 6.525769543485454, 7.294180282214006, 7.628754487387273, 8.337484955452514, 8.800302772004546, 9.164108471683325, 9.630853948648409, 10.21167590334557, 10.73775309544425, 11.00933336273420, 11.71507620643935, 12.09947899801307, 12.45482977843438, 13.23341446484385, 13.82017189390500
