| L(s) = 1 | + 2·3-s − 5-s + 4·7-s + 9-s − 4·13-s − 2·15-s + 4·19-s + 8·21-s + 6·23-s + 25-s − 4·27-s + 6·29-s − 31-s − 4·35-s + 8·37-s − 8·39-s − 6·41-s + 10·43-s − 45-s + 9·49-s + 8·57-s + 12·59-s + 14·61-s + 4·63-s + 4·65-s − 8·67-s + 12·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.10·13-s − 0.516·15-s + 0.917·19-s + 1.74·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.179·31-s − 0.676·35-s + 1.31·37-s − 1.28·39-s − 0.937·41-s + 1.52·43-s − 0.149·45-s + 9/7·49-s + 1.05·57-s + 1.56·59-s + 1.79·61-s + 0.503·63-s + 0.496·65-s − 0.977·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.971416140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.971416140\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 31 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.685114650857577692522545714306, −8.262819311601488417633173532624, −7.49419993945789631125840706374, −7.09673752484755959008347483369, −5.58426186900645716344751027443, −4.84808961466705210371002429320, −4.12976829200816980141283049639, −2.98974904852982533313874079824, −2.34845511211375451752066760608, −1.11411486849151132645867872202,
1.11411486849151132645867872202, 2.34845511211375451752066760608, 2.98974904852982533313874079824, 4.12976829200816980141283049639, 4.84808961466705210371002429320, 5.58426186900645716344751027443, 7.09673752484755959008347483369, 7.49419993945789631125840706374, 8.262819311601488417633173532624, 8.685114650857577692522545714306
