| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 5·11-s + 13-s − 15-s − 4·17-s + 3·19-s + 21-s + 3·23-s + 25-s − 27-s + 31-s − 5·33-s − 35-s − 8·37-s − 39-s − 8·41-s − 7·43-s + 45-s + 2·47-s − 6·49-s + 4·51-s − 53-s + 5·55-s − 3·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 0.258·15-s − 0.970·17-s + 0.688·19-s + 0.218·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.179·31-s − 0.870·33-s − 0.169·35-s − 1.31·37-s − 0.160·39-s − 1.24·41-s − 1.06·43-s + 0.149·45-s + 0.291·47-s − 6/7·49-s + 0.560·51-s − 0.137·53-s + 0.674·55-s − 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 31 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.94715089843126, −13.48478803419590, −13.19519928932089, −12.35960121679720, −12.18236573983719, −11.50378650769133, −11.20766589981293, −10.63748226311347, −10.01779685550544, −9.640033539742772, −9.065115739711559, −8.729729601261007, −8.115875335182025, −7.202630034141023, −6.870436844473809, −6.407628483713290, −6.069628161711604, −5.227160344789424, −4.891836551462530, −4.213351705685409, −3.487165450525752, −3.181922698338470, −2.098411142724733, −1.584182732761527, −0.9313191743600341, 0,
0.9313191743600341, 1.584182732761527, 2.098411142724733, 3.181922698338470, 3.487165450525752, 4.213351705685409, 4.891836551462530, 5.227160344789424, 6.069628161711604, 6.407628483713290, 6.870436844473809, 7.202630034141023, 8.115875335182025, 8.729729601261007, 9.065115739711559, 9.640033539742772, 10.01779685550544, 10.63748226311347, 11.20766589981293, 11.50378650769133, 12.18236573983719, 12.35960121679720, 13.19519928932089, 13.48478803419590, 13.94715089843126
