| L(s) = 1 | + 2-s + 4-s − 4·5-s − 7-s + 8-s − 4·10-s + 2·11-s − 14-s + 16-s + 17-s − 5·19-s − 4·20-s + 2·22-s + 6·23-s + 11·25-s − 28-s + 6·29-s − 31-s + 32-s + 34-s + 4·35-s − 4·37-s − 5·38-s − 4·40-s + 9·41-s − 5·43-s + 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s + 0.353·8-s − 1.26·10-s + 0.603·11-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.14·19-s − 0.894·20-s + 0.426·22-s + 1.25·23-s + 11/5·25-s − 0.188·28-s + 1.11·29-s − 0.179·31-s + 0.176·32-s + 0.171·34-s + 0.676·35-s − 0.657·37-s − 0.811·38-s − 0.632·40-s + 1.40·41-s − 0.762·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.646213914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.646213914\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.48785271151748, −12.08285939386178, −11.66416693249951, −11.34139251753498, −10.83444803655693, −10.46827085547994, −9.948517602130623, −9.153245862361781, −8.818046176853966, −8.344027328071730, −7.915763791497949, −7.284990332762818, −7.054971476128485, −6.491722724986176, −6.148065533383588, −5.395131380362569, −4.740345147218865, −4.501058424409611, −3.970277053122510, −3.558902835553664, −2.953593457628846, −2.720791591076920, −1.670475407410131, −1.086539019281341, −0.3212527876328935,
0.3212527876328935, 1.086539019281341, 1.670475407410131, 2.720791591076920, 2.953593457628846, 3.558902835553664, 3.970277053122510, 4.501058424409611, 4.740345147218865, 5.395131380362569, 6.148065533383588, 6.491722724986176, 7.054971476128485, 7.284990332762818, 7.915763791497949, 8.344027328071730, 8.818046176853966, 9.153245862361781, 9.948517602130623, 10.46827085547994, 10.83444803655693, 11.34139251753498, 11.66416693249951, 12.08285939386178, 12.48785271151748
