| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s + 3·13-s − 15-s − 7·17-s − 19-s − 21-s + 5·23-s + 25-s + 5·27-s + 5·29-s − 10·31-s + 35-s − 2·37-s − 3·39-s + 2·41-s + 6·43-s − 2·45-s − 6·49-s + 7·51-s − 9·53-s + 57-s − 7·59-s + 4·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.832·13-s − 0.258·15-s − 1.69·17-s − 0.229·19-s − 0.218·21-s + 1.04·23-s + 1/5·25-s + 0.962·27-s + 0.928·29-s − 1.79·31-s + 0.169·35-s − 0.328·37-s − 0.480·39-s + 0.312·41-s + 0.914·43-s − 0.298·45-s − 6/7·49-s + 0.980·51-s − 1.23·53-s + 0.132·57-s − 0.911·59-s + 0.512·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−7.70839715805139448605540433788, −6.74268114763584697011939134634, −6.36025544969418794968718402286, −5.56345185004455097222791799116, −4.94709938868688158912695317221, −4.19480390705796823353792434761, −3.15063118485916249534122395283, −2.27566375882933227798874081338, −1.29040153699271294413682322178, 0,
1.29040153699271294413682322178, 2.27566375882933227798874081338, 3.15063118485916249534122395283, 4.19480390705796823353792434761, 4.94709938868688158912695317221, 5.56345185004455097222791799116, 6.36025544969418794968718402286, 6.74268114763584697011939134634, 7.70839715805139448605540433788
