| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 3·11-s + 2·13-s − 2·14-s + 16-s + 3·17-s − 19-s − 3·22-s + 6·23-s − 5·25-s − 2·26-s + 2·28-s − 6·29-s − 4·31-s − 32-s − 3·34-s − 4·37-s + 38-s − 9·41-s − 43-s + 3·44-s − 6·46-s + 6·47-s − 3·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.904·11-s + 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s − 0.639·22-s + 1.25·23-s − 25-s − 0.392·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s − 0.657·37-s + 0.162·38-s − 1.40·41-s − 0.152·43-s + 0.452·44-s − 0.884·46-s + 0.875·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9335592475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9335592475\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−12.70434390882727936880586234897, −11.57619177441088474085986142761, −10.96157598355786499811507828582, −9.711604924732818951217103858923, −8.789487803962222669402730529275, −7.80670854475898716623947697687, −6.68619713403218599725876916726, −5.34383976963704470462884839275, −3.65076060485342096132488694617, −1.60693251106342657124988453680,
1.60693251106342657124988453680, 3.65076060485342096132488694617, 5.34383976963704470462884839275, 6.68619713403218599725876916726, 7.80670854475898716623947697687, 8.789487803962222669402730529275, 9.711604924732818951217103858923, 10.96157598355786499811507828582, 11.57619177441088474085986142761, 12.70434390882727936880586234897
