L(s) = 1 | + 1.47·2-s − 3-s + 0.166·4-s − 0.120·5-s − 1.47·6-s + 7-s − 2.69·8-s + 9-s − 0.177·10-s − 4.71·11-s − 0.166·12-s + 2.00·13-s + 1.47·14-s + 0.120·15-s − 4.30·16-s − 3.97·17-s + 1.47·18-s − 0.522·19-s − 0.0200·20-s − 21-s − 6.93·22-s + 6.59·23-s + 2.69·24-s − 4.98·25-s + 2.94·26-s − 27-s + 0.166·28-s + ⋯ |
L(s) = 1 | + 1.04·2-s − 0.577·3-s + 0.0831·4-s − 0.0538·5-s − 0.600·6-s + 0.377·7-s − 0.954·8-s + 0.333·9-s − 0.0560·10-s − 1.42·11-s − 0.0480·12-s + 0.555·13-s + 0.393·14-s + 0.0310·15-s − 1.07·16-s − 0.964·17-s + 0.346·18-s − 0.119·19-s − 0.00447·20-s − 0.218·21-s − 1.47·22-s + 1.37·23-s + 0.550·24-s − 0.997·25-s + 0.578·26-s − 0.192·27-s + 0.0314·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.884406057\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.884406057\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
good | 2 | \( 1 - 1.47T + 2T^{2} \) |
| 5 | \( 1 + 0.120T + 5T^{2} \) |
| 11 | \( 1 + 4.71T + 11T^{2} \) |
| 13 | \( 1 - 2.00T + 13T^{2} \) |
| 17 | \( 1 + 3.97T + 17T^{2} \) |
| 19 | \( 1 + 0.522T + 19T^{2} \) |
| 23 | \( 1 - 6.59T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 4.85T + 31T^{2} \) |
| 37 | \( 1 + 1.55T + 37T^{2} \) |
| 41 | \( 1 - 3.45T + 41T^{2} \) |
| 43 | \( 1 - 0.534T + 43T^{2} \) |
| 47 | \( 1 + 2.13T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 - 0.650T + 67T^{2} \) |
| 71 | \( 1 - 4.29T + 71T^{2} \) |
| 73 | \( 1 - 8.52T + 73T^{2} \) |
| 79 | \( 1 - 4.13T + 79T^{2} \) |
| 83 | \( 1 - 1.43T + 83T^{2} \) |
| 89 | \( 1 - 3.24T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.426075174304940039701392688012, −7.64306614101963142748824854789, −6.66422716511251972679515985635, −6.09395905148637438041587030773, −5.23000068010483783127090438135, −4.81851215820337360386597264153, −4.10517353254684964977711708642, −3.07777265868372668188266271916, −2.27036197785012768283431178978, −0.67539704673282938034090963340,
0.67539704673282938034090963340, 2.27036197785012768283431178978, 3.07777265868372668188266271916, 4.10517353254684964977711708642, 4.81851215820337360386597264153, 5.23000068010483783127090438135, 6.09395905148637438041587030773, 6.66422716511251972679515985635, 7.64306614101963142748824854789, 8.426075174304940039701392688012