| L(s) = 1 | + 2-s + 4-s + 3·5-s + 8-s + 3·10-s + 11-s + 2·13-s + 16-s − 3·17-s + 2·19-s + 3·20-s + 22-s − 3·23-s + 4·25-s + 2·26-s + 6·29-s − 4·31-s + 32-s − 3·34-s + 2·37-s + 2·38-s + 3·40-s + 3·41-s + 2·43-s + 44-s − 3·46-s + 9·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s + 0.301·11-s + 0.554·13-s + 1/4·16-s − 0.727·17-s + 0.458·19-s + 0.670·20-s + 0.213·22-s − 0.625·23-s + 4/5·25-s + 0.392·26-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s + 0.328·37-s + 0.324·38-s + 0.474·40-s + 0.468·41-s + 0.304·43-s + 0.150·44-s − 0.442·46-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.938519819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.938519819\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−7.44973255317468936062155700209, −6.81806399132678493575787700870, −6.01420467562854226082468786461, −5.86249620493476597180759611682, −4.95582870087339401900900930143, −4.26497948226297125800421724024, −3.45631936997040089244018032938, −2.52355248767095160806016081851, −1.94273181044809667918535056660, −0.988879633700524435200031686697,
0.988879633700524435200031686697, 1.94273181044809667918535056660, 2.52355248767095160806016081851, 3.45631936997040089244018032938, 4.26497948226297125800421724024, 4.95582870087339401900900930143, 5.86249620493476597180759611682, 6.01420467562854226082468786461, 6.81806399132678493575787700870, 7.44973255317468936062155700209
