| L(s) = 1 | − 3-s − 2·4-s + 9-s − 3·11-s + 2·12-s + 4·13-s + 4·16-s − 7·17-s + 3·19-s + 5·23-s − 27-s + 29-s + 31-s + 3·33-s − 2·36-s − 4·39-s − 8·41-s + 10·43-s + 6·44-s − 2·47-s − 4·48-s − 7·49-s + 7·51-s − 8·52-s + 9·53-s − 3·57-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1/3·9-s − 0.904·11-s + 0.577·12-s + 1.10·13-s + 16-s − 1.69·17-s + 0.688·19-s + 1.04·23-s − 0.192·27-s + 0.185·29-s + 0.179·31-s + 0.522·33-s − 1/3·36-s − 0.640·39-s − 1.24·41-s + 1.52·43-s + 0.904·44-s − 0.291·47-s − 0.577·48-s − 49-s + 0.980·51-s − 1.10·52-s + 1.23·53-s − 0.397·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−8.715829257419871488645667737527, −7.959803112525176802986063306805, −7.01585964897262821462622218363, −6.16739189621743937331013492260, −5.32928526515918438343438102704, −4.70629894029881924930811799521, −3.88557735981293975130711516687, −2.80527072327272521157645860615, −1.27684762329111507084674586699, 0,
1.27684762329111507084674586699, 2.80527072327272521157645860615, 3.88557735981293975130711516687, 4.70629894029881924930811799521, 5.32928526515918438343438102704, 6.16739189621743937331013492260, 7.01585964897262821462622218363, 7.959803112525176802986063306805, 8.715829257419871488645667737527
