| L(s) = 1 | − 3-s + 3·5-s + 9-s − 11-s + 3·13-s − 3·15-s − 17-s + 19-s + 7·23-s + 4·25-s − 27-s + 6·29-s − 2·31-s + 33-s − 4·37-s − 3·39-s + 9·41-s − 43-s + 3·45-s + 10·47-s − 7·49-s + 51-s − 2·53-s − 3·55-s − 57-s − 6·59-s − 12·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 0.774·15-s − 0.242·17-s + 0.229·19-s + 1.45·23-s + 4/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.174·33-s − 0.657·37-s − 0.480·39-s + 1.40·41-s − 0.152·43-s + 0.447·45-s + 1.45·47-s − 49-s + 0.140·51-s − 0.274·53-s − 0.404·55-s − 0.132·57-s − 0.781·59-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.468717283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.468717283\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−10.99020175311044249754123563552, −10.48987317087196639801806599606, −9.469292487853741982094624554205, −8.730686790656812538651452890426, −7.32843269717286178163258400168, −6.29479205975676746085811661026, −5.62818700463789751758773761734, −4.59174724472301644959370410844, −2.89883589273214610624179644709, −1.39385455464072664196875612123,
1.39385455464072664196875612123, 2.89883589273214610624179644709, 4.59174724472301644959370410844, 5.62818700463789751758773761734, 6.29479205975676746085811661026, 7.32843269717286178163258400168, 8.730686790656812538651452890426, 9.469292487853741982094624554205, 10.48987317087196639801806599606, 10.99020175311044249754123563552
