| L(s) = 1 | − 3-s + 7-s + 9-s + 2·13-s − 3·17-s + 3·19-s − 21-s − 23-s − 27-s + 6·29-s − 2·31-s − 3·37-s − 2·39-s + 3·41-s − 12·43-s + 47-s − 6·49-s + 3·51-s − 6·53-s − 3·57-s + 3·59-s − 10·61-s + 63-s + 6·67-s + 69-s + 7·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.727·17-s + 0.688·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.493·37-s − 0.320·39-s + 0.468·41-s − 1.82·43-s + 0.145·47-s − 6/7·49-s + 0.420·51-s − 0.824·53-s − 0.397·57-s + 0.390·59-s − 1.28·61-s + 0.125·63-s + 0.733·67-s + 0.120·69-s + 0.830·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−13.69289759239830, −13.10132395025154, −12.59830095697163, −12.16754203110284, −11.64000743250911, −11.12255758938180, −11.00893324642983, −10.16145000516577, −9.964727339018827, −9.254884123719161, −8.784966226824437, −8.187479839912809, −7.880974952748138, −7.108753273712830, −6.688139327300349, −6.270081850022344, −5.637238724223137, −5.122164430252840, −4.651573516919488, −4.149416576005989, −3.401192173447753, −2.957057864821220, −2.026384622843565, −1.552318765232857, −0.8232933185094097, 0,
0.8232933185094097, 1.552318765232857, 2.026384622843565, 2.957057864821220, 3.401192173447753, 4.149416576005989, 4.651573516919488, 5.122164430252840, 5.637238724223137, 6.270081850022344, 6.688139327300349, 7.108753273712830, 7.880974952748138, 8.187479839912809, 8.784966226824437, 9.254884123719161, 9.964727339018827, 10.16145000516577, 11.00893324642983, 11.12255758938180, 11.64000743250911, 12.16754203110284, 12.59830095697163, 13.10132395025154, 13.69289759239830
