| L(s) = 1 | − 1.56·2-s − 3·3-s − 5.56·4-s + 4.68·6-s + 24.2·7-s + 21.1·8-s + 9·9-s − 11·11-s + 16.6·12-s + 84.2·13-s − 37.8·14-s + 11.4·16-s − 40.9·17-s − 14.0·18-s + 120.·19-s − 72.8·21-s + 17.1·22-s − 9.94·23-s − 63.5·24-s − 131.·26-s − 27·27-s − 134.·28-s + 196.·29-s + 151.·31-s − 187.·32-s + 33·33-s + 63.9·34-s + ⋯ |
| L(s) = 1 | − 0.552·2-s − 0.577·3-s − 0.695·4-s + 0.318·6-s + 1.31·7-s + 0.935·8-s + 0.333·9-s − 0.301·11-s + 0.401·12-s + 1.79·13-s − 0.723·14-s + 0.178·16-s − 0.584·17-s − 0.184·18-s + 1.45·19-s − 0.756·21-s + 0.166·22-s − 0.0901·23-s − 0.540·24-s − 0.992·26-s − 0.192·27-s − 0.910·28-s + 1.26·29-s + 0.875·31-s − 1.03·32-s + 0.174·33-s + 0.322·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.374937157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374937157\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 1.56T + 8T^{2} \) |
| 7 | \( 1 - 24.2T + 343T^{2} \) |
| 13 | \( 1 - 84.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 120.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 9.94T + 1.21e4T^{2} \) |
| 29 | \( 1 - 196.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 253.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 179.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 90.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 483.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 491.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 127.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 628.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 87.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 390.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 165.T + 9.12e5T^{2} \) |
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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
−9.953168200527117390836305990396, −8.735525461262721975439125565720, −8.369058550036723406625319823946, −7.48891608963589717384442039059, −6.33966172896733375702714884569, −5.21844043984418814862974077306, −4.66842211640194932730591594963, −3.52338111775632256817265532671, −1.61359967550335082546408433461, −0.818194219280161136849130072491,
0.818194219280161136849130072491, 1.61359967550335082546408433461, 3.52338111775632256817265532671, 4.66842211640194932730591594963, 5.21844043984418814862974077306, 6.33966172896733375702714884569, 7.48891608963589717384442039059, 8.369058550036723406625319823946, 8.735525461262721975439125565720, 9.953168200527117390836305990396
