| L(s) = 1 | − 3.32·2-s + 3.07·4-s − 12.1·7-s + 16.3·8-s + 11·11-s + 57.2·13-s + 40.3·14-s − 79.1·16-s − 22.7·17-s − 46.2·19-s − 36.6·22-s + 128.·23-s − 190.·26-s − 37.2·28-s − 71.6·29-s − 88.7·31-s + 132.·32-s + 75.8·34-s + 30.5·37-s + 153.·38-s − 223.·41-s + 170.·43-s + 33.8·44-s − 427.·46-s − 247.·47-s − 196.·49-s + 175.·52-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.384·4-s − 0.653·7-s + 0.724·8-s + 0.301·11-s + 1.22·13-s + 0.769·14-s − 1.23·16-s − 0.325·17-s − 0.558·19-s − 0.354·22-s + 1.16·23-s − 1.43·26-s − 0.251·28-s − 0.458·29-s − 0.513·31-s + 0.730·32-s + 0.382·34-s + 0.135·37-s + 0.656·38-s − 0.850·41-s + 0.604·43-s + 0.115·44-s − 1.37·46-s − 0.768·47-s − 0.572·49-s + 0.469·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 3.32T + 8T^{2} \) |
| 7 | \( 1 + 12.1T + 343T^{2} \) |
| 13 | \( 1 - 57.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 71.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 88.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 30.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 223.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 170.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 247.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 76.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 258.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 97.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 278.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 292.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 482.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 93.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 60.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 662.T + 9.12e5T^{2} \) |
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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
−8.486937077547141786838722122946, −7.57147050166917858269938452208, −6.78514567808568869294290411727, −6.18034292829293469585193274308, −5.05534344902555634868012756813, −4.07194873907092758170597379053, −3.22010143916648247899870935062, −1.93040174555278052723866704238, −1.01363591285126632282785386679, 0,
1.01363591285126632282785386679, 1.93040174555278052723866704238, 3.22010143916648247899870935062, 4.07194873907092758170597379053, 5.05534344902555634868012756813, 6.18034292829293469585193274308, 6.78514567808568869294290411727, 7.57147050166917858269938452208, 8.486937077547141786838722122946
