| L(s) = 1 | + 3-s − 5-s + 3·7-s + 9-s − 3·11-s − 4·13-s − 15-s + 19-s + 3·21-s + 4·23-s + 25-s + 27-s + 3·29-s + 8·31-s − 3·33-s − 3·35-s − 3·37-s − 4·39-s − 5·41-s − 8·43-s − 45-s − 9·47-s + 2·49-s − 7·53-s + 3·55-s + 57-s − 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s − 0.258·15-s + 0.229·19-s + 0.654·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.557·29-s + 1.43·31-s − 0.522·33-s − 0.507·35-s − 0.493·37-s − 0.640·39-s − 0.780·41-s − 1.21·43-s − 0.149·45-s − 1.31·47-s + 2/7·49-s − 0.961·53-s + 0.404·55-s + 0.132·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.94310624461460, −15.40968374036256, −15.07512186975964, −14.55524741588766, −13.93737694219807, −13.53698525101783, −12.77308438966482, −12.26285693192401, −11.68953443654999, −11.15308406749444, −10.49394827207639, −9.966140200840387, −9.371338651972188, −8.531765439405709, −8.065613962208627, −7.798057877112287, −7.015952982242438, −6.475483190971579, −5.288597526616989, −4.849362177083357, −4.530524389674700, −3.362958205195643, −2.867826398038598, −2.060520389560167, −1.241535557426066, 0,
1.241535557426066, 2.060520389560167, 2.867826398038598, 3.362958205195643, 4.530524389674700, 4.849362177083357, 5.288597526616989, 6.475483190971579, 7.015952982242438, 7.798057877112287, 8.065613962208627, 8.531765439405709, 9.371338651972188, 9.966140200840387, 10.49394827207639, 11.15308406749444, 11.68953443654999, 12.26285693192401, 12.77308438966482, 13.53698525101783, 13.93737694219807, 14.55524741588766, 15.07512186975964, 15.40968374036256, 15.94310624461460
