| L(s) = 1 | + 3·5-s + 3·11-s + 6·13-s − 2·17-s + 2·19-s − 23-s + 4·25-s − 29-s − 5·31-s + 2·37-s + 6·41-s + 10·43-s + 2·47-s + 5·53-s + 9·55-s − 3·59-s − 2·61-s + 18·65-s − 14·67-s + 12·71-s + 6·73-s + 79-s + 15·83-s − 6·85-s − 4·89-s + 6·95-s + 9·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.904·11-s + 1.66·13-s − 0.485·17-s + 0.458·19-s − 0.208·23-s + 4/5·25-s − 0.185·29-s − 0.898·31-s + 0.328·37-s + 0.937·41-s + 1.52·43-s + 0.291·47-s + 0.686·53-s + 1.21·55-s − 0.390·59-s − 0.256·61-s + 2.23·65-s − 1.71·67-s + 1.42·71-s + 0.702·73-s + 0.112·79-s + 1.64·83-s − 0.650·85-s − 0.423·89-s + 0.615·95-s + 0.913·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.779424862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.779424862\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−13.96410852521766, −13.51642732835397, −13.12716554541431, −12.61141482466504, −12.00644426956499, −11.41516621662148, −10.84352913272480, −10.66123301592125, −9.878680162717332, −9.333163932568012, −9.050863229885003, −8.679168895059767, −7.828260714772095, −7.348883902818938, −6.552396290001693, −6.192885435142492, −5.840329115781744, −5.301788096154633, −4.490759646387811, −3.882826451786761, −3.425767116803320, −2.527938588340274, −1.971820050792112, −1.327392407813857, −0.7626102585162257,
0.7626102585162257, 1.327392407813857, 1.971820050792112, 2.527938588340274, 3.425767116803320, 3.882826451786761, 4.490759646387811, 5.301788096154633, 5.840329115781744, 6.192885435142492, 6.552396290001693, 7.348883902818938, 7.828260714772095, 8.679168895059767, 9.050863229885003, 9.333163932568012, 9.878680162717332, 10.66123301592125, 10.84352913272480, 11.41516621662148, 12.00644426956499, 12.61141482466504, 13.12716554541431, 13.51642732835397, 13.96410852521766
