| L(s) = 1 | − 3·7-s − 5·13-s + 5·19-s − 4·23-s − 4·29-s − 5·31-s − 10·37-s + 10·41-s + 43-s + 2·47-s + 2·49-s + 10·53-s + 10·59-s − 5·61-s − 3·67-s + 10·71-s − 10·73-s + 14·83-s − 16·89-s + 15·91-s − 5·97-s − 2·101-s + 16·103-s − 18·107-s + 5·109-s + 20·113-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 1.38·13-s + 1.14·19-s − 0.834·23-s − 0.742·29-s − 0.898·31-s − 1.64·37-s + 1.56·41-s + 0.152·43-s + 0.291·47-s + 2/7·49-s + 1.37·53-s + 1.30·59-s − 0.640·61-s − 0.366·67-s + 1.18·71-s − 1.17·73-s + 1.53·83-s − 1.69·89-s + 1.57·91-s − 0.507·97-s − 0.199·101-s + 1.57·103-s − 1.74·107-s + 0.478·109-s + 1.88·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059323739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059323739\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−7.67361090414951989277316178689, −7.27826903196968126667904362253, −6.64674661076555846543167715514, −5.68217025309635733156934929896, −5.31252053934765487070652491521, −4.23727742089387765140466820922, −3.52439593198683076920852242713, −2.75877704866021223073103188009, −1.92143103919424562128405886506, −0.49622159452317765602289034206,
0.49622159452317765602289034206, 1.92143103919424562128405886506, 2.75877704866021223073103188009, 3.52439593198683076920852242713, 4.23727742089387765140466820922, 5.31252053934765487070652491521, 5.68217025309635733156934929896, 6.64674661076555846543167715514, 7.27826903196968126667904362253, 7.67361090414951989277316178689
