| L(s) = 1 | − 2·3-s + 2·5-s − 4·7-s + 9-s + 2·11-s − 2·13-s − 4·15-s + 2·19-s + 8·21-s + 4·23-s − 25-s + 4·27-s − 6·29-s − 4·33-s − 8·35-s + 10·37-s + 4·39-s + 6·41-s + 6·43-s + 2·45-s + 8·47-s + 9·49-s + 6·53-s + 4·55-s − 4·57-s + 14·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 1.03·15-s + 0.458·19-s + 1.74·21-s + 0.834·23-s − 1/5·25-s + 0.769·27-s − 1.11·29-s − 0.696·33-s − 1.35·35-s + 1.64·37-s + 0.640·39-s + 0.937·41-s + 0.914·43-s + 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.824·53-s + 0.539·55-s − 0.529·57-s + 1.82·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.332662312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.332662312\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.72566147032229, −14.44601341616590, −13.71402580397446, −13.12447353111420, −12.81264623851238, −12.35078603804189, −11.64652019369567, −11.27798855750144, −10.68132929212738, −9.895117337686521, −9.798240003917594, −9.177346395212567, −8.717794538174104, −7.581404334634324, −7.118467798960168, −6.547697687393342, −6.036925772700611, −5.648072609907771, −5.212155897113223, −4.240924260842169, −3.725721240434973, −2.743463938215240, −2.361827742602419, −1.126785817064051, −0.5253789302079340,
0.5253789302079340, 1.126785817064051, 2.361827742602419, 2.743463938215240, 3.725721240434973, 4.240924260842169, 5.212155897113223, 5.648072609907771, 6.036925772700611, 6.547697687393342, 7.118467798960168, 7.581404334634324, 8.717794538174104, 9.177346395212567, 9.798240003917594, 9.895117337686521, 10.68132929212738, 11.27798855750144, 11.64652019369567, 12.35078603804189, 12.81264623851238, 13.12447353111420, 13.71402580397446, 14.44601341616590, 14.72566147032229
