| L(s) = 1 | − 3-s − 2·7-s + 9-s − 11-s + 6·13-s + 3·17-s − 19-s + 2·21-s − 5·25-s − 27-s − 6·29-s − 4·31-s + 33-s − 4·37-s − 6·39-s − 5·41-s + 12·47-s − 3·49-s − 3·51-s + 6·53-s + 57-s − 59-s − 10·61-s − 2·63-s + 3·67-s − 6·71-s + 9·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s + 0.727·17-s − 0.229·19-s + 0.436·21-s − 25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.174·33-s − 0.657·37-s − 0.960·39-s − 0.780·41-s + 1.75·47-s − 3/7·49-s − 0.420·51-s + 0.824·53-s + 0.132·57-s − 0.130·59-s − 1.28·61-s − 0.251·63-s + 0.366·67-s − 0.712·71-s + 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059626580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059626580\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
| 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
−12.54311306904648, −12.08674167209727, −11.64472527838467, −11.11820909994879, −10.71679030510566, −10.42221046148616, −9.820023090479191, −9.414152617714377, −8.921986602252668, −8.475993577544241, −7.891927504522270, −7.426902867977792, −6.987870925064135, −6.307079582766924, −6.092334820532376, −5.491141278494242, −5.315801947532580, −4.408486893999858, −3.862520100470455, −3.558417135385682, −3.073859284534440, −2.201039223703085, −1.673236023187986, −1.050557647017725, −0.3052159988025434,
0.3052159988025434, 1.050557647017725, 1.673236023187986, 2.201039223703085, 3.073859284534440, 3.558417135385682, 3.862520100470455, 4.408486893999858, 5.315801947532580, 5.491141278494242, 6.092334820532376, 6.307079582766924, 6.987870925064135, 7.426902867977792, 7.891927504522270, 8.475993577544241, 8.921986602252668, 9.414152617714377, 9.820023090479191, 10.42221046148616, 10.71679030510566, 11.11820909994879, 11.64472527838467, 12.08674167209727, 12.54311306904648
