| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 11-s + 12-s + 13-s + 15-s + 16-s + 2·17-s + 18-s + 20-s + 22-s − 6·23-s + 24-s + 25-s + 26-s + 27-s − 10·29-s + 30-s + 32-s + 33-s + 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.223·20-s + 0.213·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 1.85·29-s + 0.182·30-s + 0.176·32-s + 0.174·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.30677912134905, −12.96990328098304, −12.20921585815506, −12.11292198896851, −11.43095302225805, −11.00022376717815, −10.32830033306054, −10.05819040500834, −9.532968162988171, −9.006616395074482, −8.452125723338547, −8.107094903505802, −7.384673534552915, −7.012735309404833, −6.588058855302476, −5.794528045558830, −5.533512584691180, −5.127303677891010, −4.134960297865853, −3.936999383260509, −3.471342943529820, −2.758452108953321, −2.143474419956985, −1.746216314522780, −1.064901128588493, 0,
1.064901128588493, 1.746216314522780, 2.143474419956985, 2.758452108953321, 3.471342943529820, 3.936999383260509, 4.134960297865853, 5.127303677891010, 5.533512584691180, 5.794528045558830, 6.588058855302476, 7.012735309404833, 7.384673534552915, 8.107094903505802, 8.452125723338547, 9.006616395074482, 9.532968162988171, 10.05819040500834, 10.32830033306054, 11.00022376717815, 11.43095302225805, 12.11292198896851, 12.20921585815506, 12.96990328098304, 13.30677912134905
