| L(s) = 1 | + 0.447·2-s − 7.79·4-s + 1.93·5-s + 8.14·7-s − 7.06·8-s + 0.863·10-s + 8.40·11-s + 3.64·14-s + 59.2·16-s + 52.1·17-s − 48.8·19-s − 15.0·20-s + 3.76·22-s − 88.9·23-s − 121.·25-s − 63.5·28-s − 191.·29-s − 115.·31-s + 83.0·32-s + 23.3·34-s + 15.7·35-s + 136.·37-s − 21.8·38-s − 13.6·40-s + 436.·41-s + 202.·43-s − 65.5·44-s + ⋯ |
| L(s) = 1 | + 0.158·2-s − 0.974·4-s + 0.172·5-s + 0.439·7-s − 0.312·8-s + 0.0273·10-s + 0.230·11-s + 0.0695·14-s + 0.925·16-s + 0.743·17-s − 0.589·19-s − 0.168·20-s + 0.0364·22-s − 0.806·23-s − 0.970·25-s − 0.428·28-s − 1.22·29-s − 0.667·31-s + 0.458·32-s + 0.117·34-s + 0.0759·35-s + 0.607·37-s − 0.0932·38-s − 0.0539·40-s + 1.66·41-s + 0.716·43-s − 0.224·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.447T + 8T^{2} \) |
| 5 | \( 1 - 1.93T + 125T^{2} \) |
| 7 | \( 1 - 8.14T + 343T^{2} \) |
| 11 | \( 1 - 8.40T + 1.33e3T^{2} \) |
| 17 | \( 1 - 52.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 88.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 136.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 436.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 202.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 618.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 453.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 500.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 886.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 123.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 673.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 681.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 939.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 754.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−8.766072807263789432906937830021, −7.944585491578587656740766884997, −7.27244449784593615556648500648, −5.77296479990981163294161835044, −5.64009921029944032516287959024, −4.24678647511237328495814579314, −3.90621921499643429040701461373, −2.48427927055281865006370487960, −1.25379434328718907243274242497, 0,
1.25379434328718907243274242497, 2.48427927055281865006370487960, 3.90621921499643429040701461373, 4.24678647511237328495814579314, 5.64009921029944032516287959024, 5.77296479990981163294161835044, 7.27244449784593615556648500648, 7.944585491578587656740766884997, 8.766072807263789432906937830021
