| L(s) = 1 | − 2-s + 2·3-s + 4-s − 5-s − 2·6-s − 8-s + 9-s + 10-s + 2·12-s − 2·15-s + 16-s + 6·17-s − 18-s − 20-s − 2·24-s + 25-s − 4·27-s + 6·29-s + 2·30-s + 6·31-s − 32-s − 6·34-s + 36-s − 6·37-s + 40-s + 10·43-s − 45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.577·12-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.223·20-s − 0.408·24-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.365·30-s + 1.07·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.986·37-s + 0.158·40-s + 1.52·43-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.745512794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.745512794\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−9.127667478512558051348289952730, −8.582296893979510625560048613304, −7.87167213768836461567003864988, −7.39576925667593351270508778085, −6.34047377278705752756303160174, −5.30010101492962111385908589387, −4.02249500185660479353504857361, −3.16679097008850927954172665487, −2.38624674480783686926868893331, −0.993849910737614308794184978645,
0.993849910737614308794184978645, 2.38624674480783686926868893331, 3.16679097008850927954172665487, 4.02249500185660479353504857361, 5.30010101492962111385908589387, 6.34047377278705752756303160174, 7.39576925667593351270508778085, 7.87167213768836461567003864988, 8.582296893979510625560048613304, 9.127667478512558051348289952730
