L(s) = 1 | − 2-s + 0.882·3-s + 4-s + 5-s − 0.882·6-s + 0.390·7-s − 8-s − 2.22·9-s − 10-s − 11-s + 0.882·12-s − 0.473·13-s − 0.390·14-s + 0.882·15-s + 16-s + 3.02·17-s + 2.22·18-s − 2.37·19-s + 20-s + 0.344·21-s + 22-s + 2.87·23-s − 0.882·24-s + 25-s + 0.473·26-s − 4.60·27-s + 0.390·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.509·3-s + 0.5·4-s + 0.447·5-s − 0.360·6-s + 0.147·7-s − 0.353·8-s − 0.740·9-s − 0.316·10-s − 0.301·11-s + 0.254·12-s − 0.131·13-s − 0.104·14-s + 0.227·15-s + 0.250·16-s + 0.734·17-s + 0.523·18-s − 0.544·19-s + 0.223·20-s + 0.0751·21-s + 0.213·22-s + 0.598·23-s − 0.180·24-s + 0.200·25-s + 0.0928·26-s − 0.886·27-s + 0.0737·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
good | 3 | \( 1 - 0.882T + 3T^{2} \) |
| 7 | \( 1 - 0.390T + 7T^{2} \) |
| 13 | \( 1 + 0.473T + 13T^{2} \) |
| 17 | \( 1 - 3.02T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 - 2.87T + 23T^{2} \) |
| 29 | \( 1 - 0.244T + 29T^{2} \) |
| 31 | \( 1 - 1.27T + 31T^{2} \) |
| 37 | \( 1 + 7.22T + 37T^{2} \) |
| 41 | \( 1 + 0.0515T + 41T^{2} \) |
| 43 | \( 1 - 3.42T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 + 3.47T + 53T^{2} \) |
| 59 | \( 1 + 2.93T + 59T^{2} \) |
| 61 | \( 1 - 4.36T + 61T^{2} \) |
| 67 | \( 1 + 1.73T + 67T^{2} \) |
| 71 | \( 1 - 1.51T + 71T^{2} \) |
| 79 | \( 1 + 8.54T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 + 9.59T + 89T^{2} \) |
| 97 | \( 1 - 19.6T + 97T^{2} \) |
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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
−7.69703567850880575129206139390, −6.91414017326819815042730435322, −6.18568007628712502131550214277, −5.48920891265796047799784698351, −4.80112435571109445752878432742, −3.61263675165942453348652111842, −2.92627728661700255386501682717, −2.21251161401874244472290456146, −1.31100055865281995257273272078, 0,
1.31100055865281995257273272078, 2.21251161401874244472290456146, 2.92627728661700255386501682717, 3.61263675165942453348652111842, 4.80112435571109445752878432742, 5.48920891265796047799784698351, 6.18568007628712502131550214277, 6.91414017326819815042730435322, 7.69703567850880575129206139390