L(s) = 1 | + 2·5-s − 7-s − 4·13-s + 4·17-s − 4·23-s − 25-s − 6·29-s − 10·31-s − 2·35-s − 6·37-s + 4·41-s + 12·43-s − 10·47-s + 49-s + 6·53-s + 2·59-s − 8·65-s − 8·67-s − 12·71-s + 8·73-s + 8·79-s + 8·85-s + 6·89-s + 4·91-s − 10·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s − 1.10·13-s + 0.970·17-s − 0.834·23-s − 1/5·25-s − 1.11·29-s − 1.79·31-s − 0.338·35-s − 0.986·37-s + 0.624·41-s + 1.82·43-s − 1.45·47-s + 1/7·49-s + 0.824·53-s + 0.260·59-s − 0.992·65-s − 0.977·67-s − 1.42·71-s + 0.936·73-s + 0.900·79-s + 0.867·85-s + 0.635·89-s + 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.247579443\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.247579443\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.58823274401294, −13.05372930160472, −12.52774704760843, −12.28437775831172, −11.68082553011375, −11.08860230650485, −10.48593559794217, −10.14913673190319, −9.575233738989206, −9.297269801759120, −8.882982317700233, −7.892048347900923, −7.691968090417678, −7.115446760945763, −6.556854267070791, −5.883490019695996, −5.467926638490421, −5.245096337195723, −4.263593967009892, −3.821023922255736, −3.157826790636280, −2.475946765407266, −1.951560602986564, −1.405285792454076, −0.3220632055849756,
0.3220632055849756, 1.405285792454076, 1.951560602986564, 2.475946765407266, 3.157826790636280, 3.821023922255736, 4.263593967009892, 5.245096337195723, 5.467926638490421, 5.883490019695996, 6.556854267070791, 7.115446760945763, 7.691968090417678, 7.892048347900923, 8.882982317700233, 9.297269801759120, 9.575233738989206, 10.14913673190319, 10.48593559794217, 11.08860230650485, 11.68082553011375, 12.28437775831172, 12.52774704760843, 13.05372930160472, 13.58823274401294