| L(s) = 1 | − 5-s − 7-s − 3·9-s + 13-s − 2·17-s + 6·19-s − 3·23-s − 4·25-s − 3·29-s + 2·31-s + 35-s + 6·37-s − 3·41-s + 7·43-s + 3·45-s − 6·49-s − 4·53-s − 3·59-s + 61-s + 3·63-s − 65-s + 13·67-s + 10·71-s + 11·73-s − 8·79-s + 9·81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s + 0.277·13-s − 0.485·17-s + 1.37·19-s − 0.625·23-s − 4/5·25-s − 0.557·29-s + 0.359·31-s + 0.169·35-s + 0.986·37-s − 0.468·41-s + 1.06·43-s + 0.447·45-s − 6/7·49-s − 0.549·53-s − 0.390·59-s + 0.128·61-s + 0.377·63-s − 0.124·65-s + 1.58·67-s + 1.18·71-s + 1.28·73-s − 0.900·79-s + 81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−15.64458931106128, −15.27358410116205, −14.35789101986217, −14.11563123127527, −13.56998121937036, −12.93634251693162, −12.36333402071733, −11.75723438782620, −11.29012621022751, −11.00766441386022, −10.07548374846253, −9.516803745416343, −9.188791867051806, −8.196918184807736, −8.072110361939350, −7.322286215934008, −6.610756699609305, −5.998387241604915, −5.505865931636426, −4.793609428648859, −3.954103341259018, −3.449499941228501, −2.759685333117720, −2.012175832317286, −0.8946801993502631, 0,
0.8946801993502631, 2.012175832317286, 2.759685333117720, 3.449499941228501, 3.954103341259018, 4.793609428648859, 5.505865931636426, 5.998387241604915, 6.610756699609305, 7.322286215934008, 8.072110361939350, 8.196918184807736, 9.188791867051806, 9.516803745416343, 10.07548374846253, 11.00766441386022, 11.29012621022751, 11.75723438782620, 12.36333402071733, 12.93634251693162, 13.56998121937036, 14.11563123127527, 14.35789101986217, 15.27358410116205, 15.64458931106128
