| L(s) = 1 | − 3-s + 3·7-s − 2·9-s − 11-s − 4·13-s − 3·17-s + 5·19-s − 3·21-s + 4·23-s + 5·27-s + 5·29-s − 7·31-s + 33-s + 7·37-s + 4·39-s − 8·41-s − 6·43-s + 8·47-s + 2·49-s + 3·51-s − 9·53-s − 5·57-s − 13·61-s − 6·63-s − 12·67-s − 4·69-s + 3·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s − 2/3·9-s − 0.301·11-s − 1.10·13-s − 0.727·17-s + 1.14·19-s − 0.654·21-s + 0.834·23-s + 0.962·27-s + 0.928·29-s − 1.25·31-s + 0.174·33-s + 1.15·37-s + 0.640·39-s − 1.24·41-s − 0.914·43-s + 1.16·47-s + 2/7·49-s + 0.420·51-s − 1.23·53-s − 0.662·57-s − 1.66·61-s − 0.755·63-s − 1.46·67-s − 0.481·69-s + 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−7.898825223834204601914214192159, −7.35644155205454509301874584910, −6.53179410858690933907960973370, −5.62386195852038780865539286607, −4.95759270107606796602337935051, −4.63400988091673047994609759616, −3.24402023297722220482399143998, −2.43976174717620482782242112396, −1.32766603725939926535252095655, 0,
1.32766603725939926535252095655, 2.43976174717620482782242112396, 3.24402023297722220482399143998, 4.63400988091673047994609759616, 4.95759270107606796602337935051, 5.62386195852038780865539286607, 6.53179410858690933907960973370, 7.35644155205454509301874584910, 7.898825223834204601914214192159
