| L(s) = 1 | − 5-s + 7-s + 2·11-s + 4·13-s − 2·17-s + 2·19-s + 4·23-s + 25-s + 2·29-s + 6·31-s − 35-s − 6·37-s − 6·41-s + 4·43-s + 49-s − 8·53-s − 2·55-s − 10·61-s − 4·65-s + 12·67-s − 14·71-s + 4·73-s + 2·77-s + 8·79-s + 12·83-s + 2·85-s + 14·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.603·11-s + 1.10·13-s − 0.485·17-s + 0.458·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.07·31-s − 0.169·35-s − 0.986·37-s − 0.937·41-s + 0.609·43-s + 1/7·49-s − 1.09·53-s − 0.269·55-s − 1.28·61-s − 0.496·65-s + 1.46·67-s − 1.66·71-s + 0.468·73-s + 0.227·77-s + 0.900·79-s + 1.31·83-s + 0.216·85-s + 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.113961467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.113961467\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−8.261502148682755764449393683777, −7.59461665245224407701692158677, −6.69860477782180898455445328040, −6.26040497672845058474209688999, −5.19751449942230533251668781537, −4.55336185254856403110493202863, −3.70162908297532177570646261599, −3.01863592300913590292588521864, −1.76333598857204206892717362488, −0.837359600331921570911260549216,
0.837359600331921570911260549216, 1.76333598857204206892717362488, 3.01863592300913590292588521864, 3.70162908297532177570646261599, 4.55336185254856403110493202863, 5.19751449942230533251668781537, 6.26040497672845058474209688999, 6.69860477782180898455445328040, 7.59461665245224407701692158677, 8.261502148682755764449393683777
