| L(s) = 1 | − 1.79·2-s − 3·3-s − 4.78·4-s − 0.476·5-s + 5.37·6-s + 22.9·8-s + 9·9-s + 0.854·10-s + 15.9·11-s + 14.3·12-s − 13·13-s + 1.42·15-s − 2.78·16-s + 55.9·17-s − 16.1·18-s − 164.·19-s + 2.28·20-s − 28.5·22-s + 94.9·23-s − 68.7·24-s − 124.·25-s + 23.3·26-s − 27·27-s + 183.·29-s − 2.56·30-s − 48.7·31-s − 178.·32-s + ⋯ |
| L(s) = 1 | − 0.633·2-s − 0.577·3-s − 0.598·4-s − 0.0426·5-s + 0.365·6-s + 1.01·8-s + 0.333·9-s + 0.0270·10-s + 0.436·11-s + 0.345·12-s − 0.277·13-s + 0.0246·15-s − 0.0434·16-s + 0.797·17-s − 0.211·18-s − 1.98·19-s + 0.0255·20-s − 0.276·22-s + 0.861·23-s − 0.584·24-s − 0.998·25-s + 0.175·26-s − 0.192·27-s + 1.17·29-s − 0.0155·30-s − 0.282·31-s − 0.985·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 1.79T + 8T^{2} \) |
| 5 | \( 1 + 0.476T + 125T^{2} \) |
| 11 | \( 1 - 15.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 55.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 164.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 94.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 183.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 48.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 92.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 105.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 50.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 340.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 39.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 565.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 231.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 202.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 557.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 317.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 384.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 967.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 734.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.67e3T + 9.12e5T^{2} \) |
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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
−8.510696561057247135200001955266, −7.84104191954036989909916933814, −6.91401636448989498815456713851, −6.15254312378507653408779682058, −5.13651130194642516010065263713, −4.44195505852955115505960791168, −3.58455148054025165677688717540, −2.06883926695937467033380832582, −0.977228651530053637808023179995, 0,
0.977228651530053637808023179995, 2.06883926695937467033380832582, 3.58455148054025165677688717540, 4.44195505852955115505960791168, 5.13651130194642516010065263713, 6.15254312378507653408779682058, 6.91401636448989498815456713851, 7.84104191954036989909916933814, 8.510696561057247135200001955266
