| L(s) = 1 | − 36.1·2-s − 81·3-s + 796.·4-s + 898.·5-s + 2.93e3·6-s − 4.67e3·7-s − 1.02e4·8-s + 6.56e3·9-s − 3.25e4·10-s − 5.83e4·11-s − 6.45e4·12-s + 5.73e4·13-s + 1.69e5·14-s − 7.28e4·15-s − 3.54e4·16-s − 5.10e5·17-s − 2.37e5·18-s − 1.68e5·19-s + 7.16e5·20-s + 3.78e5·21-s + 2.11e6·22-s + 2.17e6·23-s + 8.33e5·24-s − 1.14e6·25-s − 2.07e6·26-s − 5.31e5·27-s − 3.72e6·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.577·3-s + 1.55·4-s + 0.643·5-s + 0.923·6-s − 0.735·7-s − 0.888·8-s + 0.333·9-s − 1.02·10-s − 1.20·11-s − 0.898·12-s + 0.556·13-s + 1.17·14-s − 0.371·15-s − 0.135·16-s − 1.48·17-s − 0.532·18-s − 0.296·19-s + 1.00·20-s + 0.424·21-s + 1.92·22-s + 1.62·23-s + 0.513·24-s − 0.586·25-s − 0.890·26-s − 0.192·27-s − 1.14·28-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
| \(L(5)\) |
\(\approx\) |
\(0.3625857366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3625857366\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 36.1T + 512T^{2} \) |
| 5 | \( 1 - 898.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.67e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.83e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.73e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.10e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.68e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.17e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.44e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.46e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.04e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 8.50e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.02e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.48e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.20e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 2.99e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.13e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.52e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.83e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 7.72e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.73e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.72e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.37e9T + 7.60e17T^{2} \) |
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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
−10.68771634111766829402772508998, −10.00602461457880856736627397106, −9.118131792425313576889325203054, −8.194870040475847878685088094799, −6.92859299217628609707420153182, −6.23484882877353193866083278128, −4.83046391494017891225801159487, −2.82020077117817166005402021709, −1.66424871030387010306484586483, −0.38586747602496350738196565114,
0.38586747602496350738196565114, 1.66424871030387010306484586483, 2.82020077117817166005402021709, 4.83046391494017891225801159487, 6.23484882877353193866083278128, 6.92859299217628609707420153182, 8.194870040475847878685088094799, 9.118131792425313576889325203054, 10.00602461457880856736627397106, 10.68771634111766829402772508998
