| L(s) = 1 | + 2.35·2-s − 3-s + 3.56·4-s + 2.62·5-s − 2.35·6-s + 7-s + 3.69·8-s + 9-s + 6.18·10-s + 3.18·11-s − 3.56·12-s + 1.86·13-s + 2.35·14-s − 2.62·15-s + 1.59·16-s + 2.35·18-s + 0.108·19-s + 9.35·20-s − 21-s + 7.51·22-s + 2.37·23-s − 3.69·24-s + 1.87·25-s + 4.40·26-s − 27-s + 3.56·28-s − 4.91·29-s + ⋯ |
| L(s) = 1 | + 1.66·2-s − 0.577·3-s + 1.78·4-s + 1.17·5-s − 0.963·6-s + 0.377·7-s + 1.30·8-s + 0.333·9-s + 1.95·10-s + 0.959·11-s − 1.02·12-s + 0.517·13-s + 0.630·14-s − 0.676·15-s + 0.398·16-s + 0.556·18-s + 0.0248·19-s + 2.09·20-s − 0.218·21-s + 1.60·22-s + 0.495·23-s − 0.755·24-s + 0.374·25-s + 0.863·26-s − 0.192·27-s + 0.674·28-s − 0.912·29-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
| \(L(1)\) |
\(\approx\) |
\(6.751933365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.751933365\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.35T + 2T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 19 | \( 1 - 0.108T + 19T^{2} \) |
| 23 | \( 1 - 2.37T + 23T^{2} \) |
| 29 | \( 1 + 4.91T + 29T^{2} \) |
| 31 | \( 1 - 8.37T + 31T^{2} \) |
| 37 | \( 1 - 7.42T + 37T^{2} \) |
| 41 | \( 1 - 4.38T + 41T^{2} \) |
| 43 | \( 1 + 6.27T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 - 7.25T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 9.64T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 6.12T + 73T^{2} \) |
| 79 | \( 1 - 4.31T + 79T^{2} \) |
| 83 | \( 1 + 3.59T + 83T^{2} \) |
| 89 | \( 1 + 4.30T + 89T^{2} \) |
| 97 | \( 1 - 3.80T + 97T^{2} \) |
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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
−7.80350870051822767917735662491, −6.82269532565358116938425843466, −6.34258985700343447375787746828, −5.88401465368974986841797161749, −5.20774435629773845870766102074, −4.55905816095687621861038635305, −3.86727254009256434726038920404, −2.94692826006698162384884733955, −2.01360197268676995932428001037, −1.21783830891736367648374325968,
1.21783830891736367648374325968, 2.01360197268676995932428001037, 2.94692826006698162384884733955, 3.86727254009256434726038920404, 4.55905816095687621861038635305, 5.20774435629773845870766102074, 5.88401465368974986841797161749, 6.34258985700343447375787746828, 6.82269532565358116938425843466, 7.80350870051822767917735662491
