| L(s) = 1 | − 2-s − 4-s − 5-s + 7-s + 3·8-s + 10-s − 6·11-s − 14-s − 16-s + 2·17-s − 4·19-s + 20-s + 6·22-s + 4·23-s + 25-s − 28-s − 2·29-s − 8·31-s − 5·32-s − 2·34-s − 35-s − 10·37-s + 4·38-s − 3·40-s − 10·41-s − 2·43-s + 6·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s + 0.316·10-s − 1.80·11-s − 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.223·20-s + 1.27·22-s + 0.834·23-s + 1/5·25-s − 0.188·28-s − 0.371·29-s − 1.43·31-s − 0.883·32-s − 0.342·34-s − 0.169·35-s − 1.64·37-s + 0.648·38-s − 0.474·40-s − 1.56·41-s − 0.304·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.98428954881830, −14.64646553492091, −13.76480940414807, −13.34115612169200, −13.11024474480324, −12.35298374168269, −11.98789761147165, −11.09378160884866, −10.70262219337065, −10.37093400558185, −9.930616633471559, −9.053637824852994, −8.661490394078769, −8.340686460950047, −7.586794769819370, −7.385200405241722, −6.770340875655464, −5.639719856210414, −5.302923707976870, −4.840407312759487, −4.117434476070947, −3.494653311627979, −2.762699500518753, −1.936636665743365, −1.278933228157290, 0, 0,
1.278933228157290, 1.936636665743365, 2.762699500518753, 3.494653311627979, 4.117434476070947, 4.840407312759487, 5.302923707976870, 5.639719856210414, 6.770340875655464, 7.385200405241722, 7.586794769819370, 8.340686460950047, 8.661490394078769, 9.053637824852994, 9.930616633471559, 10.37093400558185, 10.70262219337065, 11.09378160884866, 11.98789761147165, 12.35298374168269, 13.11024474480324, 13.34115612169200, 13.76480940414807, 14.64646553492091, 14.98428954881830
