| L(s) = 1 | − 6·17-s − 6·19-s − 23-s − 5·25-s + 6·29-s − 8·31-s + 2·37-s − 2·41-s − 8·43-s − 8·47-s + 2·53-s − 6·59-s − 12·67-s + 8·71-s + 6·73-s − 16·79-s + 6·83-s + 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.45·17-s − 1.37·19-s − 0.208·23-s − 25-s + 1.11·29-s − 1.43·31-s + 0.328·37-s − 0.312·41-s − 1.21·43-s − 1.16·47-s + 0.274·53-s − 0.781·59-s − 1.46·67-s + 0.949·71-s + 0.702·73-s − 1.80·79-s + 0.658·83-s + 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4094342975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4094342975\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.93392576358106, −13.42814570248104, −13.00924191495353, −12.60342231105394, −11.97488475640215, −11.36478715530318, −11.15091206065085, −10.31159098353366, −10.21407198415229, −9.348128757665285, −8.931804175008552, −8.446634254983751, −7.964328792300450, −7.319440725798321, −6.693858435247723, −6.309171995835294, −5.834273665514518, −4.909854421512116, −4.645250180536576, −3.924447232788955, −3.436648220407356, −2.539196027017422, −2.044625613806524, −1.443404994576519, −0.1970282781540039,
0.1970282781540039, 1.443404994576519, 2.044625613806524, 2.539196027017422, 3.436648220407356, 3.924447232788955, 4.645250180536576, 4.909854421512116, 5.834273665514518, 6.309171995835294, 6.693858435247723, 7.319440725798321, 7.964328792300450, 8.446634254983751, 8.931804175008552, 9.348128757665285, 10.21407198415229, 10.31159098353366, 11.15091206065085, 11.36478715530318, 11.97488475640215, 12.60342231105394, 13.00924191495353, 13.42814570248104, 13.93392576358106
