| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 3·9-s − 4·11-s + 2·13-s − 14-s + 16-s − 6·17-s + 3·18-s − 6·19-s + 4·22-s + 5·23-s − 2·26-s + 28-s − 10·29-s − 32-s + 6·34-s − 3·36-s + 8·37-s + 6·38-s − 6·41-s + 4·43-s − 4·44-s − 5·46-s − 3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 9-s − 1.20·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.707·18-s − 1.37·19-s + 0.852·22-s + 1.04·23-s − 0.392·26-s + 0.188·28-s − 1.85·29-s − 0.176·32-s + 1.02·34-s − 1/2·36-s + 1.31·37-s + 0.973·38-s − 0.937·41-s + 0.609·43-s − 0.603·44-s − 0.737·46-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2278762280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2278762280\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−14.61108553867520, −14.27037673931655, −13.20893636698156, −13.06362192116426, −12.80300301241755, −11.53842593919374, −11.33663181160614, −11.11827436033448, −10.41103792464868, −10.00281484559696, −9.065108571153669, −8.790596287705650, −8.410190151349530, −7.789178671778247, −7.249837977249906, −6.585228221436653, −6.018043037825347, −5.486065910066926, −4.839242141142660, −4.159452674300239, −3.373755451777424, −2.522287563513726, −2.270268102385046, −1.357567100729093, −0.1853268695902457,
0.1853268695902457, 1.357567100729093, 2.270268102385046, 2.522287563513726, 3.373755451777424, 4.159452674300239, 4.839242141142660, 5.486065910066926, 6.018043037825347, 6.585228221436653, 7.249837977249906, 7.789178671778247, 8.410190151349530, 8.790596287705650, 9.065108571153669, 10.00281484559696, 10.41103792464868, 11.11827436033448, 11.33663181160614, 11.53842593919374, 12.80300301241755, 13.06362192116426, 13.20893636698156, 14.27037673931655, 14.61108553867520
