L(s) = 1 | − 2-s + 2.44·3-s + 4-s + 3.50·5-s − 2.44·6-s + 1.21·7-s − 8-s + 2.96·9-s − 3.50·10-s + 1.41·11-s + 2.44·12-s + 4.25·13-s − 1.21·14-s + 8.54·15-s + 16-s − 6.92·17-s − 2.96·18-s + 1.70·19-s + 3.50·20-s + 2.95·21-s − 1.41·22-s − 6.96·23-s − 2.44·24-s + 7.25·25-s − 4.25·26-s − 0.0935·27-s + 1.21·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.40·3-s + 0.5·4-s + 1.56·5-s − 0.996·6-s + 0.457·7-s − 0.353·8-s + 0.987·9-s − 1.10·10-s + 0.426·11-s + 0.704·12-s + 1.18·13-s − 0.323·14-s + 2.20·15-s + 0.250·16-s − 1.67·17-s − 0.698·18-s + 0.392·19-s + 0.782·20-s + 0.645·21-s − 0.301·22-s − 1.45·23-s − 0.498·24-s + 1.45·25-s − 0.835·26-s − 0.0180·27-s + 0.228·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.510281279\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.510281279\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 5 | \( 1 - 3.50T + 5T^{2} \) |
| 7 | \( 1 - 1.21T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 4.25T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 23 | \( 1 + 6.96T + 23T^{2} \) |
| 29 | \( 1 + 4.05T + 29T^{2} \) |
| 31 | \( 1 - 6.03T + 31T^{2} \) |
| 37 | \( 1 - 7.22T + 37T^{2} \) |
| 41 | \( 1 + 0.637T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 0.459T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 - 3.19T + 67T^{2} \) |
| 71 | \( 1 + 6.99T + 71T^{2} \) |
| 73 | \( 1 + 5.46T + 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 3.97T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.713204008511379801957441249649, −8.006005532436952417308195016654, −7.13615513947861326067997019457, −6.22003558029984015173015280447, −5.84913351341481041535616711418, −4.49925002876549548992951106861, −3.65555170723225414855085862378, −2.47228527864446783417911513125, −2.10800704575933077548817310567, −1.23357066029666010461398375250,
1.23357066029666010461398375250, 2.10800704575933077548817310567, 2.47228527864446783417911513125, 3.65555170723225414855085862378, 4.49925002876549548992951106861, 5.84913351341481041535616711418, 6.22003558029984015173015280447, 7.13615513947861326067997019457, 8.006005532436952417308195016654, 8.713204008511379801957441249649