| L(s) = 1 | + 42.1·2-s − 81·3-s + 1.26e3·4-s + 1.72e3·5-s − 3.41e3·6-s − 4.21e3·7-s + 3.18e4·8-s + 6.56e3·9-s + 7.27e4·10-s + 8.17e4·11-s − 1.02e5·12-s − 4.73e4·13-s − 1.77e5·14-s − 1.39e5·15-s + 6.94e5·16-s − 6.03e4·17-s + 2.76e5·18-s + 4.88e5·19-s + 2.18e6·20-s + 3.41e5·21-s + 3.44e6·22-s − 6.64e5·23-s − 2.57e6·24-s + 1.02e6·25-s − 1.99e6·26-s − 5.31e5·27-s − 5.34e6·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 0.577·3-s + 2.47·4-s + 1.23·5-s − 1.07·6-s − 0.663·7-s + 2.74·8-s + 0.333·9-s + 2.30·10-s + 1.68·11-s − 1.42·12-s − 0.459·13-s − 1.23·14-s − 0.712·15-s + 2.64·16-s − 0.175·17-s + 0.621·18-s + 0.859·19-s + 3.05·20-s + 0.383·21-s + 3.13·22-s − 0.495·23-s − 1.58·24-s + 0.523·25-s − 0.856·26-s − 0.192·27-s − 1.64·28-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
| \(L(5)\) |
\(\approx\) |
\(8.523369498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.523369498\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 42.1T + 512T^{2} \) |
| 5 | \( 1 - 1.72e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.21e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.17e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 4.73e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.03e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.88e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 6.64e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.04e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.59e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.60e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.57e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.94e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.00e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.87e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 6.53e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.63e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 6.50e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.39e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.02e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.55e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.80e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.83e8T + 7.60e17T^{2} \) |
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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
−11.54448702678425386277384052084, −10.23747537167949332069384168780, −9.377881339594934260480253043782, −7.17201244237433697784775428637, −6.29325510496045971372403248203, −5.83269565968385529030897827549, −4.68372903775587009480055473514, −3.62315649699084667444914926605, −2.37250589135166338323347220297, −1.25164313540877634627582641624,
1.25164313540877634627582641624, 2.37250589135166338323347220297, 3.62315649699084667444914926605, 4.68372903775587009480055473514, 5.83269565968385529030897827549, 6.29325510496045971372403248203, 7.17201244237433697784775428637, 9.377881339594934260480253043782, 10.23747537167949332069384168780, 11.54448702678425386277384052084
