L(s) = 1 | + 2.28·2-s + 2.13·3-s + 3.22·4-s + 5-s + 4.88·6-s + 7-s + 2.80·8-s + 1.56·9-s + 2.28·10-s − 3.22·11-s + 6.90·12-s − 1.60·13-s + 2.28·14-s + 2.13·15-s − 0.0344·16-s − 2.87·17-s + 3.58·18-s − 4.80·19-s + 3.22·20-s + 2.13·21-s − 7.37·22-s + 23-s + 6.00·24-s + 25-s − 3.66·26-s − 3.05·27-s + 3.22·28-s + ⋯ |
L(s) = 1 | + 1.61·2-s + 1.23·3-s + 1.61·4-s + 0.447·5-s + 1.99·6-s + 0.377·7-s + 0.993·8-s + 0.522·9-s + 0.723·10-s − 0.972·11-s + 1.99·12-s − 0.443·13-s + 0.611·14-s + 0.551·15-s − 0.00860·16-s − 0.698·17-s + 0.845·18-s − 1.10·19-s + 0.721·20-s + 0.466·21-s − 1.57·22-s + 0.208·23-s + 1.22·24-s + 0.200·25-s − 0.717·26-s − 0.588·27-s + 0.610·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.346566241\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.346566241\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 3 | \( 1 - 2.13T + 3T^{2} \) |
| 11 | \( 1 + 3.22T + 11T^{2} \) |
| 13 | \( 1 + 1.60T + 13T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 + 4.80T + 19T^{2} \) |
| 29 | \( 1 - 1.87T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 - 1.21T + 37T^{2} \) |
| 41 | \( 1 - 2.67T + 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 6.12T + 53T^{2} \) |
| 59 | \( 1 - 5.88T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 + 1.68T + 67T^{2} \) |
| 71 | \( 1 - 5.28T + 71T^{2} \) |
| 73 | \( 1 - 8.29T + 73T^{2} \) |
| 79 | \( 1 + 7.14T + 79T^{2} \) |
| 83 | \( 1 - 9.49T + 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 - 1.29T + 97T^{2} \) |
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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
−10.44836718538708392984682108305, −9.303448889180275849289546220857, −8.484054239996756042875540807537, −7.60174216875301158482904790860, −6.60240875063218797167365824166, −5.61812010858313796644893808786, −4.71707694719561254360816839066, −3.88429300991266842873394715745, −2.62023972880161697637113742668, −2.29524834698336206983187561584,
2.29524834698336206983187561584, 2.62023972880161697637113742668, 3.88429300991266842873394715745, 4.71707694719561254360816839066, 5.61812010858313796644893808786, 6.60240875063218797167365824166, 7.60174216875301158482904790860, 8.484054239996756042875540807537, 9.303448889180275849289546220857, 10.44836718538708392984682108305