| L(s) = 1 | − 3-s − 2·7-s − 2·9-s − 11-s + 4·13-s − 6·17-s + 8·19-s + 2·21-s + 3·23-s + 5·27-s + 5·31-s + 33-s + 37-s − 4·39-s + 10·43-s − 3·49-s + 6·51-s + 6·53-s − 8·57-s + 3·59-s − 4·61-s + 4·63-s + 67-s − 3·69-s + 15·71-s + 4·73-s + 2·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s − 0.301·11-s + 1.10·13-s − 1.45·17-s + 1.83·19-s + 0.436·21-s + 0.625·23-s + 0.962·27-s + 0.898·31-s + 0.174·33-s + 0.164·37-s − 0.640·39-s + 1.52·43-s − 3/7·49-s + 0.840·51-s + 0.824·53-s − 1.05·57-s + 0.390·59-s − 0.512·61-s + 0.503·63-s + 0.122·67-s − 0.361·69-s + 1.78·71-s + 0.468·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.079546278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079546278\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−9.828597367466048724363215268486, −9.062308306545389100350755701537, −8.323465712127201941198713831607, −7.19880969537039357464113305235, −6.37561722404072890002203233082, −5.71468598548784473589814335416, −4.78230154364361768976603326542, −3.54794291769774557092878092602, −2.62441060321725601358270028596, −0.809727221508988329410874951537,
0.809727221508988329410874951537, 2.62441060321725601358270028596, 3.54794291769774557092878092602, 4.78230154364361768976603326542, 5.71468598548784473589814335416, 6.37561722404072890002203233082, 7.19880969537039357464113305235, 8.323465712127201941198713831607, 9.062308306545389100350755701537, 9.828597367466048724363215268486
