| L(s) = 1 | − 5-s − 2·7-s − 3·9-s − 11-s − 7·17-s + 4·19-s − 6·23-s − 4·25-s − 7·29-s + 2·31-s + 2·35-s − 5·37-s + 5·41-s − 8·43-s + 3·45-s + 4·47-s − 3·49-s − 53-s + 55-s + 6·59-s − 11·61-s + 6·63-s − 4·67-s + 14·71-s − 3·73-s + 2·77-s + 10·79-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 9-s − 0.301·11-s − 1.69·17-s + 0.917·19-s − 1.25·23-s − 4/5·25-s − 1.29·29-s + 0.359·31-s + 0.338·35-s − 0.821·37-s + 0.780·41-s − 1.21·43-s + 0.447·45-s + 0.583·47-s − 3/7·49-s − 0.137·53-s + 0.134·55-s + 0.781·59-s − 1.40·61-s + 0.755·63-s − 0.488·67-s + 1.66·71-s − 0.351·73-s + 0.227·77-s + 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.75134930965118, −13.51967248514548, −12.79043127033138, −12.38593452553784, −11.81108805164127, −11.38601858571476, −11.07452039217195, −10.43067557026322, −9.872116867817854, −9.370434717015208, −8.975231961606447, −8.392353935549834, −7.877958879281246, −7.475089003084208, −6.762017022152125, −6.309628272858646, −5.831990788825347, −5.268694523788541, −4.677863052595514, −3.926736935857294, −3.563114760808016, −2.945056577925459, −2.258798348223434, −1.785530769333684, −0.5154513361678598, 0,
0.5154513361678598, 1.785530769333684, 2.258798348223434, 2.945056577925459, 3.563114760808016, 3.926736935857294, 4.677863052595514, 5.268694523788541, 5.831990788825347, 6.309628272858646, 6.762017022152125, 7.475089003084208, 7.877958879281246, 8.392353935549834, 8.975231961606447, 9.370434717015208, 9.872116867817854, 10.43067557026322, 11.07452039217195, 11.38601858571476, 11.81108805164127, 12.38593452553784, 12.79043127033138, 13.51967248514548, 13.75134930965118
