| L(s) = 1 | + 3-s + 7-s − 2·9-s − 11-s + 6·13-s + 7·17-s + 19-s + 21-s + 8·23-s − 5·27-s + 6·29-s − 4·31-s − 33-s + 8·37-s + 6·39-s − 5·41-s + 6·47-s + 49-s + 7·51-s + 4·53-s + 57-s − 4·59-s − 6·61-s − 2·63-s + 5·67-s + 8·69-s − 14·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.301·11-s + 1.66·13-s + 1.69·17-s + 0.229·19-s + 0.218·21-s + 1.66·23-s − 0.962·27-s + 1.11·29-s − 0.718·31-s − 0.174·33-s + 1.31·37-s + 0.960·39-s − 0.780·41-s + 0.875·47-s + 1/7·49-s + 0.980·51-s + 0.549·53-s + 0.132·57-s − 0.520·59-s − 0.768·61-s − 0.251·63-s + 0.610·67-s + 0.963·69-s − 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.270331717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.270331717\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−16.46022250686664, −15.91953034631212, −15.22216338563871, −14.69777861940570, −14.27471523234728, −13.60993965855052, −13.23748789484120, −12.52606318423317, −11.74650938249045, −11.31978806352562, −10.67330010177425, −10.12953864043542, −9.261926726286609, −8.734496171001895, −8.317097998239946, −7.641538283400986, −7.074653865778963, −5.967875038825560, −5.725037343153548, −4.868489629179266, −3.991495312272945, −3.121298553378769, −2.897485751873712, −1.571796066760860, −0.8832350978904027,
0.8832350978904027, 1.571796066760860, 2.897485751873712, 3.121298553378769, 3.991495312272945, 4.868489629179266, 5.725037343153548, 5.967875038825560, 7.074653865778963, 7.641538283400986, 8.317097998239946, 8.734496171001895, 9.261926726286609, 10.12953864043542, 10.67330010177425, 11.31978806352562, 11.74650938249045, 12.52606318423317, 13.23748789484120, 13.60993965855052, 14.27471523234728, 14.69777861940570, 15.22216338563871, 15.91953034631212, 16.46022250686664
