| L(s) = 1 | − 5-s + 2·11-s + 4·13-s + 25-s − 2·29-s + 10·31-s + 4·37-s − 6·41-s − 4·43-s − 7·49-s − 2·53-s − 2·55-s + 6·59-s + 6·61-s − 4·65-s − 4·67-s − 8·71-s + 14·73-s + 2·79-s + 4·83-s − 14·89-s − 6·97-s + 18·101-s − 4·103-s + 16·107-s + 10·109-s − 8·113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s + 1.10·13-s + 1/5·25-s − 0.371·29-s + 1.79·31-s + 0.657·37-s − 0.937·41-s − 0.609·43-s − 49-s − 0.274·53-s − 0.269·55-s + 0.781·59-s + 0.768·61-s − 0.496·65-s − 0.488·67-s − 0.949·71-s + 1.63·73-s + 0.225·79-s + 0.439·83-s − 1.48·89-s − 0.609·97-s + 1.79·101-s − 0.394·103-s + 1.54·107-s + 0.957·109-s − 0.752·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.970312860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.970312860\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−8.312676356068141486887747723637, −7.42477295172740950822190830872, −6.57549138477247304254310444403, −6.17923449566841364294949347305, −5.17880530731220745694279259573, −4.38414527146797782079381855411, −3.68154824825393064849181438739, −2.95445518016292024291572856518, −1.74961901348703542348284567589, −0.77475453833202687945740653824,
0.77475453833202687945740653824, 1.74961901348703542348284567589, 2.95445518016292024291572856518, 3.68154824825393064849181438739, 4.38414527146797782079381855411, 5.17880530731220745694279259573, 6.17923449566841364294949347305, 6.57549138477247304254310444403, 7.42477295172740950822190830872, 8.312676356068141486887747723637
