| L(s) = 1 | − 3-s − 5-s + 9-s + 15-s − 6·17-s − 4·19-s + 25-s − 27-s + 2·29-s − 2·37-s + 2·41-s − 4·43-s − 45-s + 4·47-s − 7·49-s + 6·51-s + 10·53-s + 4·57-s − 8·59-s + 2·61-s + 4·67-s − 12·71-s + 6·73-s − 75-s + 81-s + 16·83-s + 6·85-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.149·45-s + 0.583·47-s − 49-s + 0.840·51-s + 1.37·53-s + 0.529·57-s − 1.04·59-s + 0.256·61-s + 0.488·67-s − 1.42·71-s + 0.702·73-s − 0.115·75-s + 1/9·81-s + 1.75·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7998427152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7998427152\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.22705227563026, −12.80250274496409, −12.21475571716731, −11.87365599216026, −11.33632705429509, −10.92558218138337, −10.50662700642573, −10.10320725055235, −9.368850531603748, −8.889540375233351, −8.536810565785303, −7.910234696166808, −7.404551825947852, −6.814885269526668, −6.438056392342374, −6.018625904808688, −5.281887651198447, −4.715150590723215, −4.399800724913978, −3.779159080949750, −3.202849278393593, −2.336219770030015, −1.974959307217695, −1.052440248475394, −0.3000909736046546,
0.3000909736046546, 1.052440248475394, 1.974959307217695, 2.336219770030015, 3.202849278393593, 3.779159080949750, 4.399800724913978, 4.715150590723215, 5.281887651198447, 6.018625904808688, 6.438056392342374, 6.814885269526668, 7.404551825947852, 7.910234696166808, 8.536810565785303, 8.889540375233351, 9.368850531603748, 10.10320725055235, 10.50662700642573, 10.92558218138337, 11.33632705429509, 11.87365599216026, 12.21475571716731, 12.80250274496409, 13.22705227563026
