| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 3·11-s − 15-s + 6·17-s − 2·19-s − 2·21-s + 3·23-s + 25-s − 27-s − 3·29-s + 5·31-s − 3·33-s + 2·35-s + 7·37-s + 6·41-s + 43-s + 45-s − 3·47-s − 3·49-s − 6·51-s + 6·53-s + 3·55-s + 2·57-s + 9·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.904·11-s − 0.258·15-s + 1.45·17-s − 0.458·19-s − 0.436·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s − 0.557·29-s + 0.898·31-s − 0.522·33-s + 0.338·35-s + 1.15·37-s + 0.937·41-s + 0.152·43-s + 0.149·45-s − 0.437·47-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.404·55-s + 0.264·57-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.812823947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.812823947\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.07900954031109, −12.88801810358090, −12.16393389441440, −11.86248056614573, −11.36981658821181, −10.99150761950332, −10.41020397490686, −9.970085072877248, −9.463312469034305, −9.069613738007468, −8.371030024032695, −7.971890780314157, −7.380607590952398, −6.937364999265252, −6.248923869470124, −5.927451250560138, −5.409829158250302, −4.819601073357155, −4.379860673174151, −3.780468389684228, −3.127573056840262, −2.421037259838949, −1.723809712584065, −1.131841090873799, −0.6796604722161250,
0.6796604722161250, 1.131841090873799, 1.723809712584065, 2.421037259838949, 3.127573056840262, 3.780468389684228, 4.379860673174151, 4.819601073357155, 5.409829158250302, 5.927451250560138, 6.248923869470124, 6.937364999265252, 7.380607590952398, 7.971890780314157, 8.371030024032695, 9.069613738007468, 9.463312469034305, 9.970085072877248, 10.41020397490686, 10.99150761950332, 11.36981658821181, 11.86248056614573, 12.16393389441440, 12.88801810358090, 13.07900954031109
