| L(s) = 1 | + 2.58·2-s + 4.69·4-s − 2.28·5-s + 6.97·8-s − 5.90·10-s − 2.95·11-s − 4.26·13-s + 8.64·16-s + 1.52·17-s − 7.38·19-s − 10.7·20-s − 7.64·22-s − 6.15·23-s + 0.213·25-s − 11.0·26-s − 2.34·29-s − 6.22·31-s + 8.43·32-s + 3.95·34-s + 7.17·37-s − 19.0·38-s − 15.9·40-s + 7.89·41-s + 0.834·43-s − 13.8·44-s − 15.9·46-s − 5.82·47-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 2.34·4-s − 1.02·5-s + 2.46·8-s − 1.86·10-s − 0.891·11-s − 1.18·13-s + 2.16·16-s + 0.370·17-s − 1.69·19-s − 2.39·20-s − 1.63·22-s − 1.28·23-s + 0.0426·25-s − 2.16·26-s − 0.434·29-s − 1.11·31-s + 1.49·32-s + 0.678·34-s + 1.17·37-s − 3.09·38-s − 2.51·40-s + 1.23·41-s + 0.127·43-s − 2.09·44-s − 2.34·46-s − 0.849·47-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 | \( 1 - 2.58T + 2T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 11 | \( 1 + 2.95T + 11T^{2} \) |
| 13 | \( 1 + 4.26T + 13T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 + 7.38T + 19T^{2} \) |
| 23 | \( 1 + 6.15T + 23T^{2} \) |
| 29 | \( 1 + 2.34T + 29T^{2} \) |
| 31 | \( 1 + 6.22T + 31T^{2} \) |
| 37 | \( 1 - 7.17T + 37T^{2} \) |
| 41 | \( 1 - 7.89T + 41T^{2} \) |
| 43 | \( 1 - 0.834T + 43T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 - 7.43T + 53T^{2} \) |
| 59 | \( 1 - 4.62T + 59T^{2} \) |
| 61 | \( 1 - 7.13T + 61T^{2} \) |
| 67 | \( 1 + 3.32T + 67T^{2} \) |
| 71 | \( 1 - 0.160T + 71T^{2} \) |
| 73 | \( 1 - 0.380T + 73T^{2} \) |
| 79 | \( 1 + 7.95T + 79T^{2} \) |
| 83 | \( 1 + 4.29T + 83T^{2} \) |
| 89 | \( 1 - 6.05T + 89T^{2} \) |
| 97 | \( 1 - 1.32T + 97T^{2} \) |
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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
−7.63440063268465235046508628680, −7.39560501273237445945199307530, −6.36629956253338696894121001475, −5.69426468398087684771948255366, −4.94429378728008617167483059487, −4.17788748141847368361330553909, −3.78999402105560421071422302891, −2.65569175137904122411847693929, −2.10391962718519277549757946198, 0,
2.10391962718519277549757946198, 2.65569175137904122411847693929, 3.78999402105560421071422302891, 4.17788748141847368361330553909, 4.94429378728008617167483059487, 5.69426468398087684771948255366, 6.36629956253338696894121001475, 7.39560501273237445945199307530, 7.63440063268465235046508628680
