L(s) = 1 | + 0.618·2-s − 1.61·4-s + 2.28·5-s − 2.23·8-s + 1.41·10-s − 11-s − 4.57·13-s + 1.85·16-s − 3.16·17-s + 4.03·19-s − 3.70·20-s − 0.618·22-s + 7.23·23-s + 0.236·25-s − 2.82·26-s − 1.23·29-s + 5.99·31-s + 5.61·32-s − 1.95·34-s − 10.7·37-s + 2.49·38-s − 5.11·40-s − 5.78·41-s + 5.94·43-s + 1.61·44-s + 4.47·46-s − 3.03·47-s + ⋯ |
L(s) = 1 | + 0.437·2-s − 0.809·4-s + 1.02·5-s − 0.790·8-s + 0.447·10-s − 0.301·11-s − 1.26·13-s + 0.463·16-s − 0.766·17-s + 0.925·19-s − 0.827·20-s − 0.131·22-s + 1.50·23-s + 0.0472·25-s − 0.554·26-s − 0.229·29-s + 1.07·31-s + 0.993·32-s − 0.335·34-s − 1.76·37-s + 0.404·38-s − 0.809·40-s − 0.903·41-s + 0.906·43-s + 0.243·44-s + 0.659·46-s − 0.442·47-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 | \( 1 - 0.618T + 2T^{2} \) |
| 5 | \( 1 - 2.28T + 5T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 4.57T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 - 5.99T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 5.78T + 41T^{2} \) |
| 43 | \( 1 - 5.94T + 43T^{2} \) |
| 47 | \( 1 + 3.03T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 - 1.47T + 67T^{2} \) |
| 71 | \( 1 + 5.47T + 71T^{2} \) |
| 73 | \( 1 - 6.73T + 73T^{2} \) |
| 79 | \( 1 + 7.76T + 79T^{2} \) |
| 83 | \( 1 + 8.61T + 83T^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 + 6.19T + 97T^{2} \) |
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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
−8.135587361010819300592537203845, −7.28545986894010484099735989008, −6.49888412391409346109967251294, −5.66927508676049119015465663751, −4.97257222144476557887819455931, −4.63007547215530624146475634314, −3.31133431721467592308520451539, −2.66455995315241967121597814553, −1.49319079467808264052890179337, 0,
1.49319079467808264052890179337, 2.66455995315241967121597814553, 3.31133431721467592308520451539, 4.63007547215530624146475634314, 4.97257222144476557887819455931, 5.66927508676049119015465663751, 6.49888412391409346109967251294, 7.28545986894010484099735989008, 8.135587361010819300592537203845