| L(s) = 1 | + 4·7-s − 4·11-s − 2·13-s − 17-s − 8·19-s + 10·29-s + 10·31-s + 8·37-s + 2·41-s + 4·43-s − 6·47-s + 9·49-s + 6·53-s + 8·59-s − 2·61-s − 4·67-s − 12·73-s − 16·77-s − 10·79-s + 6·83-s − 8·89-s − 8·91-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.20·11-s − 0.554·13-s − 0.242·17-s − 1.83·19-s + 1.85·29-s + 1.79·31-s + 1.31·37-s + 0.312·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s + 0.824·53-s + 1.04·59-s − 0.256·61-s − 0.488·67-s − 1.40·73-s − 1.82·77-s − 1.12·79-s + 0.658·83-s − 0.847·89-s − 0.838·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.502072125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.502072125\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.34937577609794, −13.24396373172980, −12.53077161481355, −12.11811180468496, −11.50634786999316, −11.21749244639949, −10.56435040702281, −10.20420307278605, −9.929853466413737, −8.880120392611796, −8.536415443855135, −8.201137942650025, −7.692921337932297, −7.265274837694810, −6.395495029066631, −6.167500815179910, −5.322818880482883, −4.825709889463697, −4.460229507841924, −4.118987934137518, −2.872892968379254, −2.567443878839890, −2.059710666802725, −1.219703429854836, −0.5003855195676510,
0.5003855195676510, 1.219703429854836, 2.059710666802725, 2.567443878839890, 2.872892968379254, 4.118987934137518, 4.460229507841924, 4.825709889463697, 5.322818880482883, 6.167500815179910, 6.395495029066631, 7.265274837694810, 7.692921337932297, 8.201137942650025, 8.536415443855135, 8.880120392611796, 9.929853466413737, 10.20420307278605, 10.56435040702281, 11.21749244639949, 11.50634786999316, 12.11811180468496, 12.53077161481355, 13.24396373172980, 13.34937577609794
