| L(s) = 1 | + 2·2-s − 7.96·3-s + 4·4-s + 0.313·5-s − 15.9·6-s + 8·8-s + 36.4·9-s + 0.626·10-s − 34.3·11-s − 31.8·12-s + 78.0·13-s − 2.49·15-s + 16·16-s + 19.1·17-s + 72.9·18-s − 8.01·19-s + 1.25·20-s − 68.7·22-s − 23·23-s − 63.7·24-s − 124.·25-s + 156.·26-s − 75.6·27-s − 18.7·29-s − 4.98·30-s + 337.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.53·3-s + 0.5·4-s + 0.0279·5-s − 1.08·6-s + 0.353·8-s + 1.35·9-s + 0.0197·10-s − 0.942·11-s − 0.766·12-s + 1.66·13-s − 0.0429·15-s + 0.250·16-s + 0.273·17-s + 0.955·18-s − 0.0967·19-s + 0.0139·20-s − 0.666·22-s − 0.208·23-s − 0.542·24-s − 0.999·25-s + 1.17·26-s − 0.539·27-s − 0.119·29-s − 0.0303·30-s + 1.95·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.858087545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.858087545\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 7.96T + 27T^{2} \) |
| 5 | \( 1 - 0.313T + 125T^{2} \) |
| 11 | \( 1 + 34.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 8.01T + 6.85e3T^{2} \) |
| 29 | \( 1 + 18.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 337.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 65.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 120.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 152.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 319.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 557.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 498.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 760.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 924.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 739.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 2.73T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−8.473359244625525282515306873673, −7.77491485600321516751244256746, −6.71802953885601475359581650504, −6.10770985518493945545448513241, −5.62877196414350928479964185314, −4.83004182436480186713229704789, −4.04452500048344194343275690523, −3.01016568835177532950672508757, −1.65263568856932782846263040282, −0.61129792310205010968160012969,
0.61129792310205010968160012969, 1.65263568856932782846263040282, 3.01016568835177532950672508757, 4.04452500048344194343275690523, 4.83004182436480186713229704789, 5.62877196414350928479964185314, 6.10770985518493945545448513241, 6.71802953885601475359581650504, 7.77491485600321516751244256746, 8.473359244625525282515306873673
