| L(s) = 1 | + 5-s + 4·7-s − 11-s − 4·13-s − 4·17-s − 4·19-s + 6·23-s + 25-s − 10·29-s − 4·31-s + 4·35-s − 2·37-s + 10·41-s − 8·43-s − 6·47-s + 9·49-s + 2·53-s − 55-s + 4·59-s + 10·61-s − 4·65-s − 2·67-s + 12·73-s − 4·77-s − 8·79-s − 4·85-s + 10·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 0.301·11-s − 1.10·13-s − 0.970·17-s − 0.917·19-s + 1.25·23-s + 1/5·25-s − 1.85·29-s − 0.718·31-s + 0.676·35-s − 0.328·37-s + 1.56·41-s − 1.21·43-s − 0.875·47-s + 9/7·49-s + 0.274·53-s − 0.134·55-s + 0.520·59-s + 1.28·61-s − 0.496·65-s − 0.244·67-s + 1.40·73-s − 0.455·77-s − 0.900·79-s − 0.433·85-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.228579716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.228579716\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−14.98967017694166, −14.63169230034219, −14.18795953285603, −13.40567977302406, −12.92385814658260, −12.63845199956718, −11.71280650635241, −11.22072940349014, −10.97761661186902, −10.34103913290050, −9.634717365049078, −9.096695875346162, −8.582659250769938, −8.009303999080687, −7.336776894525954, −6.994920250264955, −6.182633360391737, −5.321716515346306, −5.098259074330408, −4.464013407415887, −3.798402230901785, −2.772507229748219, −2.039828134135601, −1.765445788261497, −0.5427414738500303,
0.5427414738500303, 1.765445788261497, 2.039828134135601, 2.772507229748219, 3.798402230901785, 4.464013407415887, 5.098259074330408, 5.321716515346306, 6.182633360391737, 6.994920250264955, 7.336776894525954, 8.009303999080687, 8.582659250769938, 9.096695875346162, 9.634717365049078, 10.34103913290050, 10.97761661186902, 11.22072940349014, 11.71280650635241, 12.63845199956718, 12.92385814658260, 13.40567977302406, 14.18795953285603, 14.63169230034219, 14.98967017694166
