| L(s) = 1 | + 1.74·2-s + 1.05·4-s + 2.68·5-s − 0.749·7-s − 1.65·8-s + 4.68·10-s + 11-s − 5.19·13-s − 1.30·14-s − 4.99·16-s + 3.02·17-s − 4.44·19-s + 2.82·20-s + 1.74·22-s − 1.89·23-s + 2.20·25-s − 9.07·26-s − 0.787·28-s − 8.18·29-s + 2.38·31-s − 5.41·32-s + 5.28·34-s − 2.01·35-s + 1.94·37-s − 7.75·38-s − 4.44·40-s − 4.36·41-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.525·4-s + 1.20·5-s − 0.283·7-s − 0.586·8-s + 1.48·10-s + 0.301·11-s − 1.44·13-s − 0.349·14-s − 1.24·16-s + 0.734·17-s − 1.01·19-s + 0.630·20-s + 0.372·22-s − 0.395·23-s + 0.440·25-s − 1.78·26-s − 0.148·28-s − 1.51·29-s + 0.427·31-s − 0.957·32-s + 0.907·34-s − 0.339·35-s + 0.320·37-s − 1.25·38-s − 0.703·40-s − 0.681·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.74T + 2T^{2} \) |
| 5 | \( 1 - 2.68T + 5T^{2} \) |
| 7 | \( 1 + 0.749T + 7T^{2} \) |
| 13 | \( 1 + 5.19T + 13T^{2} \) |
| 17 | \( 1 - 3.02T + 17T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 + 8.18T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 - 1.94T + 37T^{2} \) |
| 41 | \( 1 + 4.36T + 41T^{2} \) |
| 43 | \( 1 + 6.39T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 4.94T + 53T^{2} \) |
| 59 | \( 1 - 5.51T + 59T^{2} \) |
| 67 | \( 1 - 7.88T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 6.48T + 73T^{2} \) |
| 79 | \( 1 + 5.06T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 1.72T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.51036627090034646845891038334, −6.60018322927710872339128898763, −6.23358948666351583963156137411, −5.36501812659168998465006631997, −5.03345865668244129297808223012, −4.10185877913313233437360067390, −3.32374073633396392471520549307, −2.44361680749133517566395781969, −1.78749699965979590543039916086, 0,
1.78749699965979590543039916086, 2.44361680749133517566395781969, 3.32374073633396392471520549307, 4.10185877913313233437360067390, 5.03345865668244129297808223012, 5.36501812659168998465006631997, 6.23358948666351583963156137411, 6.60018322927710872339128898763, 7.51036627090034646845891038334
