| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 6·11-s − 2·13-s − 15-s + 6·19-s − 21-s + 4·23-s + 25-s − 27-s + 4·31-s − 6·33-s + 35-s − 6·37-s + 2·39-s + 6·41-s + 4·43-s + 45-s − 2·47-s + 49-s − 53-s + 6·55-s − 6·57-s + 10·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s − 0.258·15-s + 1.37·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.718·31-s − 1.04·33-s + 0.169·35-s − 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s − 0.291·47-s + 1/7·49-s − 0.137·53-s + 0.809·55-s − 0.794·57-s + 1.30·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 356160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.358246665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.358246665\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 53 | \( 1 + T \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−12.44995399565273, −11.99514187018632, −11.59617165752887, −11.34091928248005, −10.81035961051667, −10.21258991807014, −9.783915172033509, −9.391116938329180, −9.031343290987745, −8.503286974950464, −7.910221514501357, −7.263520713453040, −6.984413209679571, −6.541343536066633, −6.005766634967847, −5.509061448055940, −5.028322723717213, −4.639140766536382, −3.971035122178963, −3.555879833739768, −2.896980366376454, −2.216370840138855, −1.614225978922389, −0.9945268506588916, −0.6877046804360860,
0.6877046804360860, 0.9945268506588916, 1.614225978922389, 2.216370840138855, 2.896980366376454, 3.555879833739768, 3.971035122178963, 4.639140766536382, 5.028322723717213, 5.509061448055940, 6.005766634967847, 6.541343536066633, 6.984413209679571, 7.263520713453040, 7.910221514501357, 8.503286974950464, 9.031343290987745, 9.391116938329180, 9.783915172033509, 10.21258991807014, 10.81035961051667, 11.34091928248005, 11.59617165752887, 11.99514187018632, 12.44995399565273
