| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s + 3·11-s + 13-s + 4·14-s + 16-s + 2·19-s + 3·22-s − 9·23-s + 26-s + 4·28-s − 6·29-s − 10·31-s + 32-s + 7·37-s + 2·38-s + 6·41-s + 10·43-s + 3·44-s − 9·46-s + 9·47-s + 9·49-s + 52-s + 6·53-s + 4·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 0.904·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.458·19-s + 0.639·22-s − 1.87·23-s + 0.196·26-s + 0.755·28-s − 1.11·29-s − 1.79·31-s + 0.176·32-s + 1.15·37-s + 0.324·38-s + 0.937·41-s + 1.52·43-s + 0.452·44-s − 1.32·46-s + 1.31·47-s + 9/7·49-s + 0.138·52-s + 0.824·53-s + 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.215354407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.215354407\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−9.552123385268823908989357476562, −8.770413921931365899679657230999, −7.74584625963531454457647124527, −7.32308184489870229838514539905, −5.96562690254234433093500390097, −5.53572937340660236773412310793, −4.28149575498690048421035953624, −3.90559286987612569172787964649, −2.32224453223285765336342353297, −1.39127211474381031402532958457,
1.39127211474381031402532958457, 2.32224453223285765336342353297, 3.90559286987612569172787964649, 4.28149575498690048421035953624, 5.53572937340660236773412310793, 5.96562690254234433093500390097, 7.32308184489870229838514539905, 7.74584625963531454457647124527, 8.770413921931365899679657230999, 9.552123385268823908989357476562
