| L(s) = 1 | − 5-s − 3·7-s − 13-s − 3·17-s + 2·19-s − 5·23-s + 25-s − 6·29-s + 10·31-s + 3·35-s + 5·37-s + 3·41-s − 4·43-s − 6·47-s + 2·49-s − 5·53-s + 8·59-s − 61-s + 65-s + 12·67-s − 71-s + 10·73-s + 79-s + 3·85-s − 89-s + 3·91-s − 2·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 0.277·13-s − 0.727·17-s + 0.458·19-s − 1.04·23-s + 1/5·25-s − 1.11·29-s + 1.79·31-s + 0.507·35-s + 0.821·37-s + 0.468·41-s − 0.609·43-s − 0.875·47-s + 2/7·49-s − 0.686·53-s + 1.04·59-s − 0.128·61-s + 0.124·65-s + 1.46·67-s − 0.118·71-s + 1.17·73-s + 0.112·79-s + 0.325·85-s − 0.105·89-s + 0.314·91-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.004457822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.004457822\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
| 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
−12.81240942034798, −12.30050236836117, −11.77561350705392, −11.46193648787452, −10.96784455457380, −10.38380578655101, −9.836068180799120, −9.589486488512793, −9.232979140888783, −8.405579728658387, −8.138932970089956, −7.680645259970629, −6.965801185588583, −6.676995686314924, −6.203965734105001, −5.715903752516752, −5.065281438614396, −4.529321732620218, −4.006977326005771, −3.523671061786997, −2.996270312130414, −2.436041164768209, −1.874703997145558, −0.9500089095470326, −0.3121165702996759,
0.3121165702996759, 0.9500089095470326, 1.874703997145558, 2.436041164768209, 2.996270312130414, 3.523671061786997, 4.006977326005771, 4.529321732620218, 5.065281438614396, 5.715903752516752, 6.203965734105001, 6.676995686314924, 6.965801185588583, 7.680645259970629, 8.138932970089956, 8.405579728658387, 9.232979140888783, 9.589486488512793, 9.836068180799120, 10.38380578655101, 10.96784455457380, 11.46193648787452, 11.77561350705392, 12.30050236836117, 12.81240942034798
