| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 3·11-s + 6·13-s + 4·14-s + 16-s − 4·17-s − 20-s + 3·22-s + 23-s − 4·25-s + 6·26-s + 4·28-s + 2·29-s − 3·31-s + 32-s − 4·34-s − 4·35-s − 7·37-s − 40-s − 5·41-s + 4·43-s + 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 0.904·11-s + 1.66·13-s + 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.223·20-s + 0.639·22-s + 0.208·23-s − 4/5·25-s + 1.17·26-s + 0.755·28-s + 0.371·29-s − 0.538·31-s + 0.176·32-s − 0.685·34-s − 0.676·35-s − 1.15·37-s − 0.158·40-s − 0.780·41-s + 0.609·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.846450745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.846450745\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.47825381732960, −12.98874200034113, −12.23544264154319, −11.91014703457970, −11.36367683801640, −11.19039603727181, −10.74047929495528, −10.22241654854304, −9.353123919089702, −8.841389347621714, −8.477102696977823, −8.017439006015348, −7.511997220252333, −6.781924270460736, −6.533149665727705, −5.823498394576666, −5.363429253367037, −4.715598237435243, −4.318620806598154, −3.722275201713352, −3.482177608596973, −2.496665465209184, −1.696597381422288, −1.539541379826193, −0.6299939433534463,
0.6299939433534463, 1.539541379826193, 1.696597381422288, 2.496665465209184, 3.482177608596973, 3.722275201713352, 4.318620806598154, 4.715598237435243, 5.363429253367037, 5.823498394576666, 6.533149665727705, 6.781924270460736, 7.511997220252333, 8.017439006015348, 8.477102696977823, 8.841389347621714, 9.353123919089702, 10.22241654854304, 10.74047929495528, 11.19039603727181, 11.36367683801640, 11.91014703457970, 12.23544264154319, 12.98874200034113, 13.47825381732960
