| L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·10-s + 4·11-s − 14-s + 16-s + 17-s + 19-s − 2·20-s + 4·22-s + 4·23-s − 25-s − 28-s − 29-s − 31-s + 32-s + 34-s + 2·35-s + 10·37-s + 38-s − 2·40-s − 5·41-s + 12·43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s + 1.20·11-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.229·19-s − 0.447·20-s + 0.852·22-s + 0.834·23-s − 1/5·25-s − 0.188·28-s − 0.185·29-s − 0.179·31-s + 0.176·32-s + 0.171·34-s + 0.338·35-s + 1.64·37-s + 0.162·38-s − 0.316·40-s − 0.780·41-s + 1.82·43-s + 0.603·44-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
| \(L(1)\) |
\(\approx\) |
\(3.461106439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.461106439\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−12.51314924282399, −12.11450015781123, −11.62543857220820, −11.26143328481655, −11.01801613125306, −10.29443544972375, −9.832715585544601, −9.198502708288154, −9.058235943171484, −8.318399512909782, −7.785076676256610, −7.376222537724617, −7.060474799276032, −6.338384304068875, −6.060090758309928, −5.616915508542848, −4.680023173331753, −4.577302300297092, −3.969809328010447, −3.463046700766988, −3.112805730396311, −2.511889426676506, −1.687023400531182, −1.158362572298884, −0.4504178195772614,
0.4504178195772614, 1.158362572298884, 1.687023400531182, 2.511889426676506, 3.112805730396311, 3.463046700766988, 3.969809328010447, 4.577302300297092, 4.680023173331753, 5.616915508542848, 6.060090758309928, 6.338384304068875, 7.060474799276032, 7.376222537724617, 7.785076676256610, 8.318399512909782, 9.058235943171484, 9.198502708288154, 9.832715585544601, 10.29443544972375, 11.01801613125306, 11.26143328481655, 11.62543857220820, 12.11450015781123, 12.51314924282399
