L(s) = 1 | + 2-s − 2.04·3-s + 4-s − 5-s − 2.04·6-s + 3.38·7-s + 8-s + 1.19·9-s − 10-s + 4.04·11-s − 2.04·12-s − 13-s + 3.38·14-s + 2.04·15-s + 16-s − 0.266·17-s + 1.19·18-s + 0.644·19-s − 20-s − 6.92·21-s + 4.04·22-s − 2.41·23-s − 2.04·24-s + 25-s − 26-s + 3.69·27-s + 3.38·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.18·3-s + 0.5·4-s − 0.447·5-s − 0.836·6-s + 1.27·7-s + 0.353·8-s + 0.398·9-s − 0.316·10-s + 1.21·11-s − 0.591·12-s − 0.277·13-s + 0.903·14-s + 0.528·15-s + 0.250·16-s − 0.0645·17-s + 0.281·18-s + 0.147·19-s − 0.223·20-s − 1.51·21-s + 0.862·22-s − 0.502·23-s − 0.418·24-s + 0.200·25-s − 0.196·26-s + 0.711·27-s + 0.639·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.252654532\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.252654532\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + 2.04T + 3T^{2} \) |
| 7 | \( 1 - 3.38T + 7T^{2} \) |
| 11 | \( 1 - 4.04T + 11T^{2} \) |
| 17 | \( 1 + 0.266T + 17T^{2} \) |
| 19 | \( 1 - 0.644T + 19T^{2} \) |
| 23 | \( 1 + 2.41T + 23T^{2} \) |
| 29 | \( 1 - 3.80T + 29T^{2} \) |
| 37 | \( 1 - 7.18T + 37T^{2} \) |
| 41 | \( 1 - 9.56T + 41T^{2} \) |
| 43 | \( 1 + 3.26T + 43T^{2} \) |
| 47 | \( 1 + 5.69T + 47T^{2} \) |
| 53 | \( 1 + 4.59T + 53T^{2} \) |
| 59 | \( 1 + 7.74T + 59T^{2} \) |
| 61 | \( 1 - 1.95T + 61T^{2} \) |
| 67 | \( 1 + 0.00670T + 67T^{2} \) |
| 71 | \( 1 - 7.58T + 71T^{2} \) |
| 73 | \( 1 + 5.98T + 73T^{2} \) |
| 79 | \( 1 + 6.01T + 79T^{2} \) |
| 83 | \( 1 + 4.01T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 1.83T + 97T^{2} \) |
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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
−8.221475018276795412401597746660, −7.61128265882388214563451098406, −6.72601823729123877052561279973, −6.13904624418670932676559148342, −5.42363298047928449209451343900, −4.56483211846302605083582686370, −4.31716273595086799618041567173, −3.11829652638582775914835228166, −1.84643065643365311680470427485, −0.861313946858545281333724320516,
0.861313946858545281333724320516, 1.84643065643365311680470427485, 3.11829652638582775914835228166, 4.31716273595086799618041567173, 4.56483211846302605083582686370, 5.42363298047928449209451343900, 6.13904624418670932676559148342, 6.72601823729123877052561279973, 7.61128265882388214563451098406, 8.221475018276795412401597746660