| L(s) = 1 | − 2·5-s + 4·7-s − 2·17-s + 8·19-s + 8·23-s − 25-s + 2·29-s + 4·31-s − 8·35-s + 10·37-s + 2·41-s + 4·43-s + 12·47-s + 9·49-s − 6·53-s − 2·61-s + 8·67-s + 12·71-s − 10·73-s + 8·79-s + 4·85-s − 14·89-s − 16·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 0.485·17-s + 1.83·19-s + 1.66·23-s − 1/5·25-s + 0.371·29-s + 0.718·31-s − 1.35·35-s + 1.64·37-s + 0.312·41-s + 0.609·43-s + 1.75·47-s + 9/7·49-s − 0.824·53-s − 0.256·61-s + 0.977·67-s + 1.42·71-s − 1.17·73-s + 0.900·79-s + 0.433·85-s − 1.48·89-s − 1.64·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.933548336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.933548336\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.43733005001475, −14.94023309638196, −14.33005035037557, −13.88528945476312, −13.38005376429570, −12.54655770669082, −12.08994411115412, −11.44581245924359, −11.19085766149727, −10.80980884751232, −9.853074533905890, −9.306278583793619, −8.678162463036379, −8.081564517165861, −7.585244091859670, −7.286704186012013, −6.434615057160771, −5.542728282840770, −5.049018666395336, −4.458331668423320, −3.937386297198965, −3.007912843711285, −2.417158995185983, −1.261892973664478, −0.8021727643499916,
0.8021727643499916, 1.261892973664478, 2.417158995185983, 3.007912843711285, 3.937386297198965, 4.458331668423320, 5.049018666395336, 5.542728282840770, 6.434615057160771, 7.286704186012013, 7.585244091859670, 8.081564517165861, 8.678162463036379, 9.306278583793619, 9.853074533905890, 10.80980884751232, 11.19085766149727, 11.44581245924359, 12.08994411115412, 12.54655770669082, 13.38005376429570, 13.88528945476312, 14.33005035037557, 14.94023309638196, 15.43733005001475
