| L(s) = 1 | + 3-s − 4·7-s + 9-s + 3·11-s − 3·13-s + 17-s + 19-s − 4·21-s − 3·23-s + 27-s − 10·29-s + 6·31-s + 3·33-s + 4·37-s − 3·39-s + 5·41-s + 43-s + 2·47-s + 9·49-s + 51-s + 14·53-s + 57-s − 6·59-s + 8·61-s − 4·63-s + 12·67-s − 3·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.904·11-s − 0.832·13-s + 0.242·17-s + 0.229·19-s − 0.872·21-s − 0.625·23-s + 0.192·27-s − 1.85·29-s + 1.07·31-s + 0.522·33-s + 0.657·37-s − 0.480·39-s + 0.780·41-s + 0.152·43-s + 0.291·47-s + 9/7·49-s + 0.140·51-s + 1.92·53-s + 0.132·57-s − 0.781·59-s + 1.02·61-s − 0.503·63-s + 1.46·67-s − 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.850404124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.850404124\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
| 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
−8.244022891423895989561308872543, −7.40080586687189559809707521983, −6.89091279074149099443669476691, −6.13873412599510221461752555595, −5.45133061500081446581594646034, −4.22431790531969563742284589933, −3.72320999299337715202760871453, −2.88153473653649950737471871382, −2.09659259915980125198169433708, −0.70120583849027539965215713529,
0.70120583849027539965215713529, 2.09659259915980125198169433708, 2.88153473653649950737471871382, 3.72320999299337715202760871453, 4.22431790531969563742284589933, 5.45133061500081446581594646034, 6.13873412599510221461752555595, 6.89091279074149099443669476691, 7.40080586687189559809707521983, 8.244022891423895989561308872543
