| L(s) = 1 | + 11-s + 6·13-s + 5·17-s + 6·19-s + 23-s − 5·25-s − 9·29-s − 6·31-s + 6·37-s − 6·41-s + 6·43-s − 11·47-s − 14·53-s + 2·61-s − 10·67-s − 3·71-s + 5·73-s − 5·79-s + 8·83-s + 10·89-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s + 1.66·13-s + 1.21·17-s + 1.37·19-s + 0.208·23-s − 25-s − 1.67·29-s − 1.07·31-s + 0.986·37-s − 0.937·41-s + 0.914·43-s − 1.60·47-s − 1.92·53-s + 0.256·61-s − 1.22·67-s − 0.356·71-s + 0.585·73-s − 0.562·79-s + 0.878·83-s + 1.05·89-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.45760625171522, −13.08830841091905, −12.66416526972801, −11.88401151363191, −11.67592780037281, −11.07544144272003, −10.86326759368461, −10.07570845354132, −9.583149923856831, −9.299119766217654, −8.781968213470302, −8.031455480522122, −7.740694755733586, −7.353096928885410, −6.577596414664382, −6.044722323686115, −5.696335861816536, −5.216624253576169, −4.522037706753785, −3.732670232739507, −3.473143696348780, −3.082602533381398, −1.989396672308026, −1.489972926799600, −0.9871053527562070, 0,
0.9871053527562070, 1.489972926799600, 1.989396672308026, 3.082602533381398, 3.473143696348780, 3.732670232739507, 4.522037706753785, 5.216624253576169, 5.696335861816536, 6.044722323686115, 6.577596414664382, 7.353096928885410, 7.740694755733586, 8.031455480522122, 8.781968213470302, 9.299119766217654, 9.583149923856831, 10.07570845354132, 10.86326759368461, 11.07544144272003, 11.67592780037281, 11.88401151363191, 12.66416526972801, 13.08830841091905, 13.45760625171522
