| L(s) = 1 | + 1.56·2-s + 81·3-s − 509.·4-s − 1.26e3·5-s + 126.·6-s − 7.65e3·7-s − 1.60e3·8-s + 6.56e3·9-s − 1.97e3·10-s + 7.03e4·11-s − 4.12e4·12-s − 4.76e4·13-s − 1.19e4·14-s − 1.02e5·15-s + 2.58e5·16-s + 3.89e5·17-s + 1.02e4·18-s + 5.11e5·19-s + 6.43e5·20-s − 6.20e5·21-s + 1.10e5·22-s + 1.72e5·23-s − 1.29e5·24-s − 3.58e5·25-s − 7.46e4·26-s + 5.31e5·27-s + 3.90e6·28-s + ⋯ |
| L(s) = 1 | + 0.0692·2-s + 0.577·3-s − 0.995·4-s − 0.903·5-s + 0.0399·6-s − 1.20·7-s − 0.138·8-s + 0.333·9-s − 0.0625·10-s + 1.44·11-s − 0.574·12-s − 0.463·13-s − 0.0834·14-s − 0.521·15-s + 0.985·16-s + 1.13·17-s + 0.0230·18-s + 0.900·19-s + 0.899·20-s − 0.695·21-s + 0.100·22-s + 0.128·23-s − 0.0797·24-s − 0.183·25-s − 0.0320·26-s + 0.192·27-s + 1.19·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 - 1.56T + 512T^{2} \) |
| 5 | \( 1 + 1.26e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 7.65e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.03e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 4.76e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.89e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.11e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.72e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.84e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.87e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.06e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.23e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.93e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.33e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.91e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.21e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.99e6T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.08e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 7.68e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.66e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.97e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.74e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.38e8T + 7.60e17T^{2} \) |
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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
−10.10142803402548107049164003466, −9.488502741261435835439067928529, −8.599090168208643902781968405207, −7.55804679715634307358557024481, −6.44670102214275231378531546136, −4.93905328504460000845609328276, −3.67048252348304748837499773653, −3.29653158525177859601862824887, −1.15883047383314893714019520627, 0,
1.15883047383314893714019520627, 3.29653158525177859601862824887, 3.67048252348304748837499773653, 4.93905328504460000845609328276, 6.44670102214275231378531546136, 7.55804679715634307358557024481, 8.599090168208643902781968405207, 9.488502741261435835439067928529, 10.10142803402548107049164003466
