| L(s) = 1 | − 2.78·2-s + 0.603·3-s + 5.78·4-s + 3.11·5-s − 1.68·6-s + 0.146·7-s − 10.5·8-s − 2.63·9-s − 8.68·10-s + 5.95·11-s + 3.48·12-s − 2.13·13-s − 0.408·14-s + 1.87·15-s + 17.8·16-s − 3.51·17-s + 7.35·18-s + 2.37·19-s + 18.0·20-s + 0.0885·21-s − 16.6·22-s − 3.35·23-s − 6.36·24-s + 4.69·25-s + 5.95·26-s − 3.40·27-s + 0.847·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 0.348·3-s + 2.89·4-s + 1.39·5-s − 0.687·6-s + 0.0554·7-s − 3.72·8-s − 0.878·9-s − 2.74·10-s + 1.79·11-s + 1.00·12-s − 0.592·13-s − 0.109·14-s + 0.485·15-s + 4.46·16-s − 0.853·17-s + 1.73·18-s + 0.544·19-s + 4.02·20-s + 0.0193·21-s − 3.53·22-s − 0.699·23-s − 1.29·24-s + 0.939·25-s + 1.16·26-s − 0.654·27-s + 0.160·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 6047 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.78T + 2T^{2} \) |
| 3 | \( 1 - 0.603T + 3T^{2} \) |
| 5 | \( 1 - 3.11T + 5T^{2} \) |
| 7 | \( 1 - 0.146T + 7T^{2} \) |
| 11 | \( 1 - 5.95T + 11T^{2} \) |
| 13 | \( 1 + 2.13T + 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + 3.35T + 23T^{2} \) |
| 29 | \( 1 - 1.70T + 29T^{2} \) |
| 31 | \( 1 + 2.22T + 31T^{2} \) |
| 37 | \( 1 + 6.81T + 37T^{2} \) |
| 41 | \( 1 + 2.99T + 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 + 6.38T + 47T^{2} \) |
| 53 | \( 1 + 8.02T + 53T^{2} \) |
| 59 | \( 1 + 0.780T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 0.544T + 67T^{2} \) |
| 71 | \( 1 - 0.166T + 71T^{2} \) |
| 73 | \( 1 - 3.03T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 9.95T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−8.078225543691189195935617308134, −6.96855943745647290397386961407, −6.60273150178614402968695423943, −6.02020498258522097134372297374, −5.19006594399862153951371964053, −3.58265626233615114997832078371, −2.74013813243274036885208487847, −1.88010177578360162348155682175, −1.46970523475947229615240151887, 0,
1.46970523475947229615240151887, 1.88010177578360162348155682175, 2.74013813243274036885208487847, 3.58265626233615114997832078371, 5.19006594399862153951371964053, 6.02020498258522097134372297374, 6.60273150178614402968695423943, 6.96855943745647290397386961407, 8.078225543691189195935617308134
