| L(s) = 1 | + 2.45·2-s + 3-s + 4.04·4-s − 2.57·5-s + 2.45·6-s − 2.91·7-s + 5.03·8-s + 9-s − 6.32·10-s − 3.44·11-s + 4.04·12-s + 13-s − 7.17·14-s − 2.57·15-s + 4.28·16-s + 3.27·17-s + 2.45·18-s − 5.14·19-s − 10.4·20-s − 2.91·21-s − 8.47·22-s − 1.47·23-s + 5.03·24-s + 1.61·25-s + 2.45·26-s + 27-s − 11.8·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 0.577·3-s + 2.02·4-s − 1.15·5-s + 1.00·6-s − 1.10·7-s + 1.78·8-s + 0.333·9-s − 2.00·10-s − 1.03·11-s + 1.16·12-s + 0.277·13-s − 1.91·14-s − 0.664·15-s + 1.07·16-s + 0.793·17-s + 0.579·18-s − 1.18·19-s − 2.32·20-s − 0.636·21-s − 1.80·22-s − 0.307·23-s + 1.02·24-s + 0.323·25-s + 0.482·26-s + 0.192·27-s − 2.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 5 | \( 1 + 2.57T + 5T^{2} \) |
| 7 | \( 1 + 2.91T + 7T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 19 | \( 1 + 5.14T + 19T^{2} \) |
| 23 | \( 1 + 1.47T + 23T^{2} \) |
| 29 | \( 1 + 3.75T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 8.53T + 37T^{2} \) |
| 41 | \( 1 + 1.42T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 6.04T + 47T^{2} \) |
| 53 | \( 1 + 2.04T + 53T^{2} \) |
| 59 | \( 1 - 9.76T + 59T^{2} \) |
| 61 | \( 1 + 0.588T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 8.60T + 71T^{2} \) |
| 73 | \( 1 - 6.11T + 73T^{2} \) |
| 79 | \( 1 + 0.797T + 79T^{2} \) |
| 83 | \( 1 - 3.23T + 83T^{2} \) |
| 89 | \( 1 - 8.60T + 89T^{2} \) |
| 97 | \( 1 + 0.801T + 97T^{2} \) |
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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
−7.81073936286851520655477057454, −7.22126734348403956272254954283, −6.51306263028736798346612836567, −5.68584293011543099626400769607, −4.98820373038946084341066196226, −3.85868710625065198631291488636, −3.73410666757383817467917414789, −2.91984242137464738710457779002, −2.04997973015640340388422824718, 0,
2.04997973015640340388422824718, 2.91984242137464738710457779002, 3.73410666757383817467917414789, 3.85868710625065198631291488636, 4.98820373038946084341066196226, 5.68584293011543099626400769607, 6.51306263028736798346612836567, 7.22126734348403956272254954283, 7.81073936286851520655477057454
