| L(s) = 1 | − 5-s + 3·11-s − 13-s − 3·17-s + 5·19-s − 23-s − 4·25-s + 9·29-s + 4·31-s + 5·37-s − 7·41-s − 3·43-s + 8·47-s + 9·53-s − 3·55-s − 4·59-s − 2·61-s + 65-s − 12·67-s − 8·71-s + 13·73-s − 8·79-s − 13·83-s + 3·85-s + 9·89-s − 5·95-s + 17·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.904·11-s − 0.277·13-s − 0.727·17-s + 1.14·19-s − 0.208·23-s − 4/5·25-s + 1.67·29-s + 0.718·31-s + 0.821·37-s − 1.09·41-s − 0.457·43-s + 1.16·47-s + 1.23·53-s − 0.404·55-s − 0.520·59-s − 0.256·61-s + 0.124·65-s − 1.46·67-s − 0.949·71-s + 1.52·73-s − 0.900·79-s − 1.42·83-s + 0.325·85-s + 0.953·89-s − 0.512·95-s + 1.72·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.236989692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.236989692\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−14.16102006772377, −13.75455052011619, −13.41100381179114, −12.66766637171668, −11.97060058201567, −11.81459514597867, −11.47032689904336, −10.64889123835245, −10.10319685835278, −9.772374925197093, −8.968096917859923, −8.713327067113233, −8.036556480782758, −7.440042636706514, −7.050882472009271, −6.306184502286291, −5.999605153176435, −5.126380319125855, −4.554094713954125, −4.124649411905766, −3.401571774417317, −2.832023345361263, −2.074866740701574, −1.243040753150477, −0.5487567586613977,
0.5487567586613977, 1.243040753150477, 2.074866740701574, 2.832023345361263, 3.401571774417317, 4.124649411905766, 4.554094713954125, 5.126380319125855, 5.999605153176435, 6.306184502286291, 7.050882472009271, 7.440042636706514, 8.036556480782758, 8.713327067113233, 8.968096917859923, 9.772374925197093, 10.10319685835278, 10.64889123835245, 11.47032689904336, 11.81459514597867, 11.97060058201567, 12.66766637171668, 13.41100381179114, 13.75455052011619, 14.16102006772377
