| L(s) = 1 | + 2·5-s + 3·7-s − 3·11-s + 13-s + 8·17-s − 19-s − 3·23-s − 25-s − 4·29-s + 2·31-s + 6·35-s − 4·37-s − 43-s + 6·47-s + 2·49-s − 14·53-s − 6·55-s + 14·59-s + 10·61-s + 2·65-s − 3·67-s − 9·71-s − 6·73-s − 9·77-s + 11·79-s + 16·83-s + 16·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.13·7-s − 0.904·11-s + 0.277·13-s + 1.94·17-s − 0.229·19-s − 0.625·23-s − 1/5·25-s − 0.742·29-s + 0.359·31-s + 1.01·35-s − 0.657·37-s − 0.152·43-s + 0.875·47-s + 2/7·49-s − 1.92·53-s − 0.809·55-s + 1.82·59-s + 1.28·61-s + 0.248·65-s − 0.366·67-s − 1.06·71-s − 0.702·73-s − 1.02·77-s + 1.23·79-s + 1.75·83-s + 1.73·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.198406192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.198406192\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−15.96867047394786, −15.10110376559877, −14.68192208204705, −14.09171032103626, −13.81565365379796, −13.04919963625753, −12.61778146341983, −11.82642063746640, −11.46935606163964, −10.64901334708378, −10.18848912080703, −9.831460982318412, −9.002203797622078, −8.382254232622383, −7.678421469322581, −7.562528618053282, −6.386628452448337, −5.830528840270988, −5.278985065096443, −4.864942895556325, −3.856169154497140, −3.176571017251010, −2.171153794942957, −1.733012903926288, −0.7692540095483815,
0.7692540095483815, 1.733012903926288, 2.171153794942957, 3.176571017251010, 3.856169154497140, 4.864942895556325, 5.278985065096443, 5.830528840270988, 6.386628452448337, 7.562528618053282, 7.678421469322581, 8.382254232622383, 9.002203797622078, 9.831460982318412, 10.18848912080703, 10.64901334708378, 11.46935606163964, 11.82642063746640, 12.61778146341983, 13.04919963625753, 13.81565365379796, 14.09171032103626, 14.68192208204705, 15.10110376559877, 15.96867047394786
