| L(s) = 1 | + 2·2-s + 2.52·3-s + 4·4-s + 15.4·5-s + 5.05·6-s + 8·8-s − 20.6·9-s + 30.8·10-s + 16.2·11-s + 10.1·12-s − 50.2·13-s + 38.9·15-s + 16·16-s − 130.·17-s − 41.2·18-s − 85.0·19-s + 61.6·20-s + 32.4·22-s + 23·23-s + 20.2·24-s + 112.·25-s − 100.·26-s − 120.·27-s − 65.1·29-s + 77.8·30-s + 87.4·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.485·3-s + 0.5·4-s + 1.37·5-s + 0.343·6-s + 0.353·8-s − 0.763·9-s + 0.975·10-s + 0.445·11-s + 0.242·12-s − 1.07·13-s + 0.670·15-s + 0.250·16-s − 1.86·17-s − 0.540·18-s − 1.02·19-s + 0.689·20-s + 0.314·22-s + 0.208·23-s + 0.171·24-s + 0.902·25-s − 0.757·26-s − 0.857·27-s − 0.416·29-s + 0.473·30-s + 0.506·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 2.52T + 27T^{2} \) |
| 5 | \( 1 - 15.4T + 125T^{2} \) |
| 11 | \( 1 - 16.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 130.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 85.0T + 6.85e3T^{2} \) |
| 29 | \( 1 + 65.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 87.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 341.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 423.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 487.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 102.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 565.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 344.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 203.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 118.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 449.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 309.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 274.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 530.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 775.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.07e3T + 9.12e5T^{2} \) |
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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
−8.505116926750535387171163269641, −7.29883462053660060536548518330, −6.52092076377020047789663954465, −5.98226138148689350384203916448, −5.06713581757718677070683905932, −4.36556709995627433991798232724, −3.16774567851705146272050393467, −2.27321501754380247473395513884, −1.86344786070412688562537402386, 0,
1.86344786070412688562537402386, 2.27321501754380247473395513884, 3.16774567851705146272050393467, 4.36556709995627433991798232724, 5.06713581757718677070683905932, 5.98226138148689350384203916448, 6.52092076377020047789663954465, 7.29883462053660060536548518330, 8.505116926750535387171163269641
