| L(s) = 1 | + 3-s + 5-s − 3·7-s − 2·9-s + 11-s + 15-s + 3·17-s − 3·21-s + 4·23-s − 4·25-s − 5·27-s + 8·31-s + 33-s − 3·35-s − 7·37-s + 8·41-s + 43-s − 2·45-s + 7·47-s + 2·49-s + 3·51-s + 6·53-s + 55-s + 10·59-s + 8·61-s + 6·63-s + 8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s − 2/3·9-s + 0.301·11-s + 0.258·15-s + 0.727·17-s − 0.654·21-s + 0.834·23-s − 4/5·25-s − 0.962·27-s + 1.43·31-s + 0.174·33-s − 0.507·35-s − 1.15·37-s + 1.24·41-s + 0.152·43-s − 0.298·45-s + 1.02·47-s + 2/7·49-s + 0.420·51-s + 0.824·53-s + 0.134·55-s + 1.30·59-s + 1.02·61-s + 0.755·63-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.896995132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.896995132\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.58210430144937, −13.26050723072425, −12.62284664516628, −12.18547702083960, −11.74521184041010, −11.13978449752456, −10.56557648515156, −10.03430109048129, −9.575930158488522, −9.285767981871842, −8.709480923994336, −8.207703836443963, −7.734566335823016, −6.965366532379585, −6.667427787573512, −5.996173064699527, −5.585250992450520, −5.104159135720232, −4.182868732950739, −3.631724083600500, −3.253870083876419, −2.470213254661126, −2.280690342691475, −1.147519773343255, −0.5409172227096059,
0.5409172227096059, 1.147519773343255, 2.280690342691475, 2.470213254661126, 3.253870083876419, 3.631724083600500, 4.182868732950739, 5.104159135720232, 5.585250992450520, 5.996173064699527, 6.667427787573512, 6.965366532379585, 7.734566335823016, 8.207703836443963, 8.709480923994336, 9.285767981871842, 9.575930158488522, 10.03430109048129, 10.56557648515156, 11.13978449752456, 11.74521184041010, 12.18547702083960, 12.62284664516628, 13.26050723072425, 13.58210430144937
