| L(s) = 1 | + 1.53·2-s + 0.360·4-s + 3.15·5-s − 2.51·8-s + 4.85·10-s − 5.74·11-s − 0.360·13-s − 4.59·16-s + 2.77·17-s − 7.23·19-s + 1.13·20-s − 8.83·22-s − 0.824·23-s + 4.97·25-s − 0.554·26-s − 4.27·29-s − 4.98·31-s − 2.01·32-s + 4.26·34-s + 7.49·37-s − 11.1·38-s − 7.95·40-s − 3.33·41-s − 7.86·43-s − 2.07·44-s − 1.26·46-s − 3.48·47-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.180·4-s + 1.41·5-s − 0.890·8-s + 1.53·10-s − 1.73·11-s − 0.100·13-s − 1.14·16-s + 0.673·17-s − 1.65·19-s + 0.254·20-s − 1.88·22-s − 0.171·23-s + 0.994·25-s − 0.108·26-s − 0.794·29-s − 0.894·31-s − 0.356·32-s + 0.731·34-s + 1.23·37-s − 1.80·38-s − 1.25·40-s − 0.520·41-s − 1.20·43-s − 0.312·44-s − 0.186·46-s − 0.507·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 5 | \( 1 - 3.15T + 5T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 13 | \( 1 + 0.360T + 13T^{2} \) |
| 17 | \( 1 - 2.77T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 0.824T + 23T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 - 7.49T + 37T^{2} \) |
| 41 | \( 1 + 3.33T + 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 + 3.48T + 47T^{2} \) |
| 53 | \( 1 + 2.91T + 53T^{2} \) |
| 59 | \( 1 - 2.39T + 59T^{2} \) |
| 61 | \( 1 - 3.20T + 61T^{2} \) |
| 67 | \( 1 - 1.89T + 67T^{2} \) |
| 71 | \( 1 - 1.60T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 5.47T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.144157863142191258599936991558, −7.15757664528957792464032550717, −6.19139980599586353538077015910, −5.75046688377008109227870114205, −5.16150231440569629864531336583, −4.49028345069840469569321045550, −3.40472449264272195698221173636, −2.55727868543143765386821822517, −1.89599294907052143510114325066, 0,
1.89599294907052143510114325066, 2.55727868543143765386821822517, 3.40472449264272195698221173636, 4.49028345069840469569321045550, 5.16150231440569629864531336583, 5.75046688377008109227870114205, 6.19139980599586353538077015910, 7.15757664528957792464032550717, 8.144157863142191258599936991558
