| L(s) = 1 | − 24.5·2-s − 5.41·3-s + 91.2·4-s + 546.·5-s + 133.·6-s + 9.85e3·7-s + 1.03e4·8-s − 1.96e4·9-s − 1.34e4·10-s + 4.42e4·11-s − 494.·12-s − 2.41e5·14-s − 2.96e3·15-s − 3.00e5·16-s − 7.92e4·17-s + 4.82e5·18-s − 8.67e5·19-s + 4.98e4·20-s − 5.33e4·21-s − 1.08e6·22-s + 5.41e5·23-s − 5.60e4·24-s − 1.65e6·25-s + 2.13e5·27-s + 8.99e5·28-s + 4.30e6·29-s + 7.27e4·30-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 0.0386·3-s + 0.178·4-s + 0.390·5-s + 0.0419·6-s + 1.55·7-s + 0.891·8-s − 0.998·9-s − 0.424·10-s + 0.910·11-s − 0.00688·12-s − 1.68·14-s − 0.0150·15-s − 1.14·16-s − 0.229·17-s + 1.08·18-s − 1.52·19-s + 0.0696·20-s − 0.0598·21-s − 0.988·22-s + 0.403·23-s − 0.0344·24-s − 0.847·25-s + 0.0771·27-s + 0.276·28-s + 1.12·29-s + 0.0163·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 24.5T + 512T^{2} \) |
| 3 | \( 1 + 5.41T + 1.96e4T^{2} \) |
| 5 | \( 1 - 546.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.85e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.42e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 7.92e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.67e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.41e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.30e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.59e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.27e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 5.31e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.91e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.80e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.71e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.19e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.20e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.26e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.08e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.50e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 7.71e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.18e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.89e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.30e9T + 7.60e17T^{2} \) |
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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
−10.68300181607327125511473544652, −9.314961960650946170194375615775, −8.598987897627520060984321204150, −7.938472883489577298678843437034, −6.58334547098537131089844065544, −5.22706270977621561195561067114, −4.16169551522331016307700928709, −2.17113675551465102467618299225, −1.31772629476998820612185867648, 0,
1.31772629476998820612185867648, 2.17113675551465102467618299225, 4.16169551522331016307700928709, 5.22706270977621561195561067114, 6.58334547098537131089844065544, 7.938472883489577298678843437034, 8.598987897627520060984321204150, 9.314961960650946170194375615775, 10.68300181607327125511473544652
