| L(s) = 1 | − 2·5-s + 4·7-s − 13-s + 2·17-s + 6·19-s + 8·23-s − 25-s − 4·29-s + 4·31-s − 8·35-s + 6·37-s + 8·43-s + 9·49-s − 8·59-s − 10·61-s + 2·65-s + 14·67-s + 8·71-s − 2·73-s + 14·79-s − 4·83-s − 4·85-s − 8·89-s − 4·91-s − 12·95-s − 10·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 0.277·13-s + 0.485·17-s + 1.37·19-s + 1.66·23-s − 1/5·25-s − 0.742·29-s + 0.718·31-s − 1.35·35-s + 0.986·37-s + 1.21·43-s + 9/7·49-s − 1.04·59-s − 1.28·61-s + 0.248·65-s + 1.71·67-s + 0.949·71-s − 0.234·73-s + 1.57·79-s − 0.439·83-s − 0.433·85-s − 0.847·89-s − 0.419·91-s − 1.23·95-s − 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.550522424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.550522424\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 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.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−15.99669433459663, −15.32255037945571, −15.06458444550354, −14.44384981705328, −13.92092585079513, −13.39505914980260, −12.42877780336261, −12.16611702403745, −11.35684139169192, −11.19128968470231, −10.64592718223830, −9.559156313660945, −9.325363299403312, −8.302723912875123, −7.970541595373147, −7.450309214708669, −6.973242628919322, −5.875472653964039, −5.229477857518872, −4.729248064743017, −4.075687567420530, −3.273272966621485, −2.520571471353287, −1.436177386250942, −0.7722565586564398,
0.7722565586564398, 1.436177386250942, 2.520571471353287, 3.273272966621485, 4.075687567420530, 4.729248064743017, 5.229477857518872, 5.875472653964039, 6.973242628919322, 7.450309214708669, 7.970541595373147, 8.302723912875123, 9.325363299403312, 9.559156313660945, 10.64592718223830, 11.19128968470231, 11.35684139169192, 12.16611702403745, 12.42877780336261, 13.39505914980260, 13.92092585079513, 14.44384981705328, 15.06458444550354, 15.32255037945571, 15.99669433459663
