| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s − 2·7-s + 8-s + 6·9-s − 4·11-s − 3·12-s + 3·13-s − 2·14-s + 16-s − 17-s + 6·18-s + 3·19-s + 6·21-s − 4·22-s + 6·23-s − 3·24-s + 3·26-s − 9·27-s − 2·28-s + 9·29-s − 3·31-s + 32-s + 12·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.755·7-s + 0.353·8-s + 2·9-s − 1.20·11-s − 0.866·12-s + 0.832·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s + 1.41·18-s + 0.688·19-s + 1.30·21-s − 0.852·22-s + 1.25·23-s − 0.612·24-s + 0.588·26-s − 1.73·27-s − 0.377·28-s + 1.67·29-s − 0.538·31-s + 0.176·32-s + 2.08·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.194977691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.194977691\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−10.46570418925199429920376580209, −9.784275464323046908920314757241, −8.369885645844106625648883770383, −7.06943044381842946186852516381, −6.59981138392846208863192075593, −5.58767301501897318687391027675, −5.17464859876377144696336477541, −4.06844075693930644580697011502, −2.79262582015668274957851201942, −0.874471729084186316705636996898,
0.874471729084186316705636996898, 2.79262582015668274957851201942, 4.06844075693930644580697011502, 5.17464859876377144696336477541, 5.58767301501897318687391027675, 6.59981138392846208863192075593, 7.06943044381842946186852516381, 8.369885645844106625648883770383, 9.784275464323046908920314757241, 10.46570418925199429920376580209
