| L(s) = 1 | + 2·5-s − 4·11-s + 4·13-s − 4·17-s − 2·19-s + 23-s − 25-s + 6·29-s + 6·31-s − 2·37-s − 10·41-s − 8·43-s − 10·47-s + 6·53-s − 8·55-s + 12·59-s + 10·61-s + 8·65-s − 8·67-s − 4·71-s + 2·73-s + 8·79-s + 6·83-s − 8·85-s − 4·95-s + 8·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s + 1.10·13-s − 0.970·17-s − 0.458·19-s + 0.208·23-s − 1/5·25-s + 1.11·29-s + 1.07·31-s − 0.328·37-s − 1.56·41-s − 1.21·43-s − 1.45·47-s + 0.824·53-s − 1.07·55-s + 1.56·59-s + 1.28·61-s + 0.992·65-s − 0.977·67-s − 0.474·71-s + 0.234·73-s + 0.900·79-s + 0.658·83-s − 0.867·85-s − 0.410·95-s + 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.42138876624932, −13.19670892413319, −12.80539828626132, −11.99044471721502, −11.56129070085619, −11.17929028254389, −10.40482482888873, −10.14211815522131, −10.02570245261242, −8.991909360131212, −8.772648162366476, −8.200187080769118, −7.923579657061537, −6.983757590444081, −6.529591876134389, −6.312756287801118, −5.587499559711687, −5.046008304639250, −4.759604567118825, −3.928492744931925, −3.365627862364376, −2.709427994392148, −2.172031780748108, −1.667185640711948, −0.8536131433345161, 0,
0.8536131433345161, 1.667185640711948, 2.172031780748108, 2.709427994392148, 3.365627862364376, 3.928492744931925, 4.759604567118825, 5.046008304639250, 5.587499559711687, 6.312756287801118, 6.529591876134389, 6.983757590444081, 7.923579657061537, 8.200187080769118, 8.772648162366476, 8.991909360131212, 10.02570245261242, 10.14211815522131, 10.40482482888873, 11.17929028254389, 11.56129070085619, 11.99044471721502, 12.80539828626132, 13.19670892413319, 13.42138876624932
