| L(s) = 1 | + 0.244·2-s − 3-s − 1.94·4-s − 2.06·5-s − 0.244·6-s − 7-s − 0.964·8-s + 9-s − 0.506·10-s + 3.04·11-s + 1.94·12-s − 6.80·13-s − 0.244·14-s + 2.06·15-s + 3.64·16-s + 0.244·18-s − 0.484·19-s + 4.01·20-s + 21-s + 0.746·22-s + 0.437·23-s + 0.964·24-s − 0.721·25-s − 1.66·26-s − 27-s + 1.94·28-s − 8.45·29-s + ⋯ |
| L(s) = 1 | + 0.173·2-s − 0.577·3-s − 0.970·4-s − 0.925·5-s − 0.0999·6-s − 0.377·7-s − 0.340·8-s + 0.333·9-s − 0.160·10-s + 0.919·11-s + 0.560·12-s − 1.88·13-s − 0.0654·14-s + 0.534·15-s + 0.911·16-s + 0.0576·18-s − 0.111·19-s + 0.897·20-s + 0.218·21-s + 0.159·22-s + 0.0911·23-s + 0.196·24-s − 0.144·25-s − 0.326·26-s − 0.192·27-s + 0.366·28-s − 1.56·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1977175457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1977175457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.244T + 2T^{2} \) |
| 5 | \( 1 + 2.06T + 5T^{2} \) |
| 11 | \( 1 - 3.04T + 11T^{2} \) |
| 13 | \( 1 + 6.80T + 13T^{2} \) |
| 19 | \( 1 + 0.484T + 19T^{2} \) |
| 23 | \( 1 - 0.437T + 23T^{2} \) |
| 29 | \( 1 + 8.45T + 29T^{2} \) |
| 31 | \( 1 + 2.11T + 31T^{2} \) |
| 37 | \( 1 + 8.74T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 - 6.29T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 0.0579T + 53T^{2} \) |
| 59 | \( 1 + 8.36T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 2.15T + 71T^{2} \) |
| 73 | \( 1 - 6.96T + 73T^{2} \) |
| 79 | \( 1 + 2.30T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 4.63T + 97T^{2} \) |
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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
−7.991286817691019810321301195083, −7.29464973868351159722768967525, −6.79465251010246307578785738398, −5.72534500642276836320731805270, −5.16209532792952177782470437689, −4.37645616591491727199564991845, −3.87480179642909555680201877750, −3.06466824844604654341513648178, −1.69048166989844545536599263738, −0.23256967817525608074498281194,
0.23256967817525608074498281194, 1.69048166989844545536599263738, 3.06466824844604654341513648178, 3.87480179642909555680201877750, 4.37645616591491727199564991845, 5.16209532792952177782470437689, 5.72534500642276836320731805270, 6.79465251010246307578785738398, 7.29464973868351159722768967525, 7.991286817691019810321301195083
