| L(s) = 1 | + 2-s + 0.260·3-s + 4-s − 4.36·5-s + 0.260·6-s + 0.775·7-s + 8-s − 2.93·9-s − 4.36·10-s − 4.36·11-s + 0.260·12-s + 0.775·14-s − 1.13·15-s + 16-s − 6.93·17-s − 2.93·18-s − 19-s − 4.36·20-s + 0.201·21-s − 4.36·22-s − 4.08·23-s + 0.260·24-s + 14.0·25-s − 1.54·27-s + 0.775·28-s + 2.54·29-s − 1.13·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.150·3-s + 0.5·4-s − 1.95·5-s + 0.106·6-s + 0.292·7-s + 0.353·8-s − 0.977·9-s − 1.37·10-s − 1.31·11-s + 0.0751·12-s + 0.207·14-s − 0.293·15-s + 0.250·16-s − 1.68·17-s − 0.691·18-s − 0.229·19-s − 0.975·20-s + 0.0440·21-s − 0.930·22-s − 0.850·23-s + 0.0531·24-s + 2.80·25-s − 0.297·27-s + 0.146·28-s + 0.472·29-s − 0.207·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9471249962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9471249962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.260T + 3T^{2} \) |
| 5 | \( 1 + 4.36T + 5T^{2} \) |
| 7 | \( 1 - 0.775T + 7T^{2} \) |
| 11 | \( 1 + 4.36T + 11T^{2} \) |
| 17 | \( 1 + 6.93T + 17T^{2} \) |
| 23 | \( 1 + 4.08T + 23T^{2} \) |
| 29 | \( 1 - 2.54T + 29T^{2} \) |
| 31 | \( 1 - 6.41T + 31T^{2} \) |
| 37 | \( 1 + 6.64T + 37T^{2} \) |
| 41 | \( 1 - 1.20T + 41T^{2} \) |
| 43 | \( 1 - 8.33T + 43T^{2} \) |
| 47 | \( 1 + 1.36T + 47T^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 2.03T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 2.30T + 73T^{2} \) |
| 79 | \( 1 - 8.16T + 79T^{2} \) |
| 83 | \( 1 - 2.75T + 83T^{2} \) |
| 89 | \( 1 + 3.34T + 89T^{2} \) |
| 97 | \( 1 + 2.12T + 97T^{2} \) |
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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
−8.065879702989229847794172845595, −7.40930747226303658150525455424, −6.64812735644728818347146639441, −5.85466558664076881177866799537, −4.73323681889433691496252443171, −4.60320061103967595956323976820, −3.63138013638691196137149535940, −2.93416319396040820862478291700, −2.22298476268853446349101569001, −0.41705027983549294543378248442,
0.41705027983549294543378248442, 2.22298476268853446349101569001, 2.93416319396040820862478291700, 3.63138013638691196137149535940, 4.60320061103967595956323976820, 4.73323681889433691496252443171, 5.85466558664076881177866799537, 6.64812735644728818347146639441, 7.40930747226303658150525455424, 8.065879702989229847794172845595
