| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 3·11-s + 12-s + 15-s + 16-s + 3·17-s + 18-s − 2·19-s + 20-s + 3·22-s + 9·23-s + 24-s + 25-s + 27-s − 6·29-s + 30-s + 4·31-s + 32-s + 3·33-s + 3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s + 0.288·12-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.639·22-s + 1.87·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.182·30-s + 0.718·31-s + 0.176·32-s + 0.522·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.760224553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.760224553\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−12.97177951434705, −12.58764719445715, −11.94304926683683, −11.56512149423447, −11.08046167741329, −10.62829538180429, −9.855727067706813, −9.819082638096458, −9.041841802761879, −8.723904473651969, −8.227329437154043, −7.535292511399504, −7.125238034713062, −6.644819047261208, −6.271635595314986, −5.536248009393384, −5.199630147284670, −4.631283565660706, −3.978249446841416, −3.621928053889961, −2.996251055766996, −2.560302400126651, −1.848462359504109, −1.324421023121998, −0.7008765261152128,
0.7008765261152128, 1.324421023121998, 1.848462359504109, 2.560302400126651, 2.996251055766996, 3.621928053889961, 3.978249446841416, 4.631283565660706, 5.199630147284670, 5.536248009393384, 6.271635595314986, 6.644819047261208, 7.125238034713062, 7.535292511399504, 8.227329437154043, 8.723904473651969, 9.041841802761879, 9.819082638096458, 9.855727067706813, 10.62829538180429, 11.08046167741329, 11.56512149423447, 11.94304926683683, 12.58764719445715, 12.97177951434705
