| L(s) = 1 | + 4.03·2-s + 3·3-s + 8.30·4-s − 0.236·5-s + 12.1·6-s + 13.8·7-s + 1.23·8-s + 9·9-s − 0.955·10-s + 11·11-s + 24.9·12-s − 48.2·13-s + 56.0·14-s − 0.709·15-s − 61.4·16-s − 73.8·17-s + 36.3·18-s − 142.·19-s − 1.96·20-s + 41.6·21-s + 44.4·22-s + 140.·23-s + 3.71·24-s − 124.·25-s − 194.·26-s + 27·27-s + 115.·28-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 0.577·3-s + 1.03·4-s − 0.0211·5-s + 0.824·6-s + 0.749·7-s + 0.0547·8-s + 0.333·9-s − 0.0302·10-s + 0.301·11-s + 0.599·12-s − 1.02·13-s + 1.06·14-s − 0.0122·15-s − 0.960·16-s − 1.05·17-s + 0.475·18-s − 1.72·19-s − 0.0219·20-s + 0.432·21-s + 0.430·22-s + 1.27·23-s + 0.0316·24-s − 0.999·25-s − 1.46·26-s + 0.192·27-s + 0.778·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 4.03T + 8T^{2} \) |
| 5 | \( 1 + 0.236T + 125T^{2} \) |
| 7 | \( 1 - 13.8T + 343T^{2} \) |
| 13 | \( 1 + 48.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 73.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 190.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 228.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 129.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 42.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 2.80T + 1.03e5T^{2} \) |
| 53 | \( 1 - 595.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 897.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 475.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 147.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.43e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 669.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.03e3T + 9.12e5T^{2} \) |
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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
−8.528334266965220714498234435578, −7.29892252440210349534967956595, −6.84930938683964388386047866898, −5.76522257744356415191499807657, −4.98839158203143940407262170021, −4.27723548140373713709559952420, −3.66710483173031379106383123750, −2.43611075158851003639899747870, −1.92512283818076852900433249572, 0,
1.92512283818076852900433249572, 2.43611075158851003639899747870, 3.66710483173031379106383123750, 4.27723548140373713709559952420, 4.98839158203143940407262170021, 5.76522257744356415191499807657, 6.84930938683964388386047866898, 7.29892252440210349534967956595, 8.528334266965220714498234435578
