L(s) = 1 | + 2-s + 0.104·3-s + 4-s + 1.37·5-s + 0.104·6-s − 2.65·7-s + 8-s − 2.98·9-s + 1.37·10-s − 1.16·11-s + 0.104·12-s + 4.98·13-s − 2.65·14-s + 0.143·15-s + 16-s − 2.86·17-s − 2.98·18-s + 5.01·19-s + 1.37·20-s − 0.276·21-s − 1.16·22-s − 23-s + 0.104·24-s − 3.10·25-s + 4.98·26-s − 0.623·27-s − 2.65·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.0601·3-s + 0.5·4-s + 0.615·5-s + 0.0425·6-s − 1.00·7-s + 0.353·8-s − 0.996·9-s + 0.435·10-s − 0.351·11-s + 0.0300·12-s + 1.38·13-s − 0.709·14-s + 0.0370·15-s + 0.250·16-s − 0.695·17-s − 0.704·18-s + 1.15·19-s + 0.307·20-s − 0.0603·21-s − 0.248·22-s − 0.208·23-s + 0.0212·24-s − 0.621·25-s + 0.976·26-s − 0.119·27-s − 0.501·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 - 0.104T + 3T^{2} \) |
| 5 | \( 1 - 1.37T + 5T^{2} \) |
| 7 | \( 1 + 2.65T + 7T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 - 4.98T + 13T^{2} \) |
| 17 | \( 1 + 2.86T + 17T^{2} \) |
| 19 | \( 1 - 5.01T + 19T^{2} \) |
| 29 | \( 1 + 5.01T + 29T^{2} \) |
| 31 | \( 1 + 8.98T + 31T^{2} \) |
| 37 | \( 1 - 3.46T + 37T^{2} \) |
| 41 | \( 1 + 2.09T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 + 7.29T + 53T^{2} \) |
| 59 | \( 1 - 5.87T + 59T^{2} \) |
| 61 | \( 1 + 4.37T + 61T^{2} \) |
| 67 | \( 1 - 7.08T + 67T^{2} \) |
| 71 | \( 1 + 4.81T + 71T^{2} \) |
| 73 | \( 1 - 5.86T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 2.79T + 83T^{2} \) |
| 89 | \( 1 - 2.93T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−7.68197522436441066130916174984, −6.70761845815876808139521235662, −6.19259154758239780835965381145, −5.62538548840410438907287590622, −5.05994419812136038264412574822, −3.71029660998978705131653469879, −3.42980992626175207293399762807, −2.49635681448440238916137208891, −1.57867599167048096416684925027, 0,
1.57867599167048096416684925027, 2.49635681448440238916137208891, 3.42980992626175207293399762807, 3.71029660998978705131653469879, 5.05994419812136038264412574822, 5.62538548840410438907287590622, 6.19259154758239780835965381145, 6.70761845815876808139521235662, 7.68197522436441066130916174984