| L(s) = 1 | − 2-s − 2·3-s + 4-s + 5-s + 2·6-s − 8-s + 9-s − 10-s + 2·11-s − 2·12-s + 6·13-s − 2·15-s + 16-s − 18-s + 8·19-s + 20-s − 2·22-s + 2·23-s + 2·24-s + 25-s − 6·26-s + 4·27-s − 6·29-s + 2·30-s − 2·31-s − 32-s − 4·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.577·12-s + 1.66·13-s − 0.516·15-s + 1/4·16-s − 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.426·22-s + 0.417·23-s + 0.408·24-s + 1/5·25-s − 1.17·26-s + 0.769·27-s − 1.11·29-s + 0.365·30-s − 0.359·31-s − 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.62250035502200, −13.11023967659765, −12.55396010367318, −11.96940379653505, −11.64247765967087, −11.12577701531377, −10.87707165818835, −10.43623003211715, −9.665155297505744, −9.373735321471180, −8.936404814895969, −8.320702637944007, −7.805314271491378, −7.147635502598255, −6.685603026333322, −6.208662734709749, −5.810190196189840, −5.280000052751757, −4.891988962381617, −3.917373748420708, −3.396796413898017, −2.926598586488426, −1.805945423882578, −1.376523530084650, −0.8754816217628506, 0,
0.8754816217628506, 1.376523530084650, 1.805945423882578, 2.926598586488426, 3.396796413898017, 3.917373748420708, 4.891988962381617, 5.280000052751757, 5.810190196189840, 6.208662734709749, 6.685603026333322, 7.147635502598255, 7.805314271491378, 8.320702637944007, 8.936404814895969, 9.373735321471180, 9.665155297505744, 10.43623003211715, 10.87707165818835, 11.12577701531377, 11.64247765967087, 11.96940379653505, 12.55396010367318, 13.11023967659765, 13.62250035502200
