| L(s) = 1 | + 1.90·2-s + 0.182·3-s − 4.35·4-s + 17.8·5-s + 0.347·6-s − 23.5·8-s − 26.9·9-s + 34.0·10-s − 20.0·11-s − 0.794·12-s − 44.6·13-s + 3.25·15-s − 10.1·16-s − 10.1·17-s − 51.4·18-s − 51.5·19-s − 77.8·20-s − 38.3·22-s + 155.·23-s − 4.29·24-s + 194.·25-s − 85.2·26-s − 9.83·27-s − 261.·29-s + 6.21·30-s − 211.·31-s + 169.·32-s + ⋯ |
| L(s) = 1 | + 0.674·2-s + 0.0350·3-s − 0.544·4-s + 1.59·5-s + 0.0236·6-s − 1.04·8-s − 0.998·9-s + 1.07·10-s − 0.550·11-s − 0.0191·12-s − 0.952·13-s + 0.0560·15-s − 0.158·16-s − 0.145·17-s − 0.673·18-s − 0.622·19-s − 0.870·20-s − 0.371·22-s + 1.40·23-s − 0.0365·24-s + 1.55·25-s − 0.642·26-s − 0.0701·27-s − 1.67·29-s + 0.0378·30-s − 1.22·31-s + 0.935·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.425658075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.425658075\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 1.90T + 8T^{2} \) |
| 3 | \( 1 - 0.182T + 27T^{2} \) |
| 5 | \( 1 - 17.8T + 125T^{2} \) |
| 11 | \( 1 + 20.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 51.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 261.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 211.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 252.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 19.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 7.94T + 1.03e5T^{2} \) |
| 53 | \( 1 - 259.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 888.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 198.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 454.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 10.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 29.9T + 3.89e5T^{2} \) |
| 79 | \( 1 - 906.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 117.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 210.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.20e3T + 9.12e5T^{2} \) |
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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
−8.956022432477353483379718579702, −7.87750564451697265418028255514, −6.83840039574980296934452885548, −5.92306076521061975935687092805, −5.42685132270256761787823800735, −4.95731121157142174650320058931, −3.77528693558942369915712308361, −2.67646437548670705629039996787, −2.19125704750686182405654825285, −0.59387523267756943103182275173,
0.59387523267756943103182275173, 2.19125704750686182405654825285, 2.67646437548670705629039996787, 3.77528693558942369915712308361, 4.95731121157142174650320058931, 5.42685132270256761787823800735, 5.92306076521061975935687092805, 6.83840039574980296934452885548, 7.87750564451697265418028255514, 8.956022432477353483379718579702
