| L(s) = 1 | − 3-s − 2·5-s + 9-s − 5·13-s + 2·15-s + 6·17-s + 8·19-s + 23-s − 25-s − 27-s + 5·29-s − 5·31-s − 8·37-s + 5·39-s − 3·41-s − 43-s − 2·45-s + 2·47-s − 6·51-s − 10·53-s − 8·57-s − 3·59-s + 3·61-s + 10·65-s + 13·67-s − 69-s + 15·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.38·13-s + 0.516·15-s + 1.45·17-s + 1.83·19-s + 0.208·23-s − 1/5·25-s − 0.192·27-s + 0.928·29-s − 0.898·31-s − 1.31·37-s + 0.800·39-s − 0.468·41-s − 0.152·43-s − 0.298·45-s + 0.291·47-s − 0.840·51-s − 1.37·53-s − 1.05·57-s − 0.390·59-s + 0.384·61-s + 1.24·65-s + 1.58·67-s − 0.120·69-s + 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.085504400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.085504400\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−13.43220977181033, −12.51475944330044, −12.38785629118742, −12.03440310666738, −11.57843149175300, −11.13083931106169, −10.51686188639622, −10.00931819769019, −9.572534170399647, −9.284717962395587, −8.282178834408740, −7.962501955762067, −7.518877615451932, −7.021356568623281, −6.699716648798881, −5.753012087510368, −5.254163381661298, −5.094648978988110, −4.375767895206341, −3.615951327852443, −3.312813202363637, −2.661451284516858, −1.754858612777386, −1.073155137409236, −0.3679582439787542,
0.3679582439787542, 1.073155137409236, 1.754858612777386, 2.661451284516858, 3.312813202363637, 3.615951327852443, 4.375767895206341, 5.094648978988110, 5.254163381661298, 5.753012087510368, 6.699716648798881, 7.021356568623281, 7.518877615451932, 7.962501955762067, 8.282178834408740, 9.284717962395587, 9.572534170399647, 10.00931819769019, 10.51686188639622, 11.13083931106169, 11.57843149175300, 12.03440310666738, 12.38785629118742, 12.51475944330044, 13.43220977181033
