| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 6·13-s + 14-s + 16-s − 8·19-s − 6·23-s + 6·26-s − 28-s + 2·29-s + 10·31-s − 32-s + 6·37-s + 8·38-s − 4·41-s + 2·43-s + 6·46-s − 8·47-s + 49-s − 6·52-s + 56-s − 2·58-s + 12·59-s + 2·61-s − 10·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 1.83·19-s − 1.25·23-s + 1.17·26-s − 0.188·28-s + 0.371·29-s + 1.79·31-s − 0.176·32-s + 0.986·37-s + 1.29·38-s − 0.624·41-s + 0.304·43-s + 0.884·46-s − 1.16·47-s + 1/7·49-s − 0.832·52-s + 0.133·56-s − 0.262·58-s + 1.56·59-s + 0.256·61-s − 1.27·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9410506538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9410506538\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.45864732451823, −11.99185016904462, −11.55613515832848, −11.14520086415917, −10.43046379806258, −10.14729746226587, −9.765529059474637, −9.544884466719625, −8.712350663031870, −8.364535550791500, −8.045968602666762, −7.520308665337734, −6.893441375946732, −6.570909332082159, −6.173616286490678, −5.622921635938841, −4.819117906693998, −4.604476983192030, −3.944518663265072, −3.361355719183426, −2.554216847746652, −2.330214350874626, −1.868769445790557, −0.8523798123694757, −0.3410335529892947,
0.3410335529892947, 0.8523798123694757, 1.868769445790557, 2.330214350874626, 2.554216847746652, 3.361355719183426, 3.944518663265072, 4.604476983192030, 4.819117906693998, 5.622921635938841, 6.173616286490678, 6.570909332082159, 6.893441375946732, 7.520308665337734, 8.045968602666762, 8.364535550791500, 8.712350663031870, 9.544884466719625, 9.765529059474637, 10.14729746226587, 10.43046379806258, 11.14520086415917, 11.55613515832848, 11.99185016904462, 12.45864732451823
