| L(s) = 1 | − 2-s − 3-s + 4-s − 0.694·5-s + 6-s + 1.42·7-s − 8-s + 9-s + 0.694·10-s − 0.355·11-s − 12-s − 3.56·13-s − 1.42·14-s + 0.694·15-s + 16-s + 1.04·17-s − 18-s + 5.27·19-s − 0.694·20-s − 1.42·21-s + 0.355·22-s + 23-s + 24-s − 4.51·25-s + 3.56·26-s − 27-s + 1.42·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.310·5-s + 0.408·6-s + 0.537·7-s − 0.353·8-s + 0.333·9-s + 0.219·10-s − 0.107·11-s − 0.288·12-s − 0.987·13-s − 0.380·14-s + 0.179·15-s + 0.250·16-s + 0.254·17-s − 0.235·18-s + 1.20·19-s − 0.155·20-s − 0.310·21-s + 0.0757·22-s + 0.208·23-s + 0.204·24-s − 0.903·25-s + 0.698·26-s − 0.192·27-s + 0.268·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 0.694T + 5T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 + 0.355T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 1.04T + 17T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 31 | \( 1 + 5.90T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 41 | \( 1 - 7.61T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 - 9.84T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 8.02T + 59T^{2} \) |
| 61 | \( 1 + 5.20T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 2.57T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 4.12T + 79T^{2} \) |
| 83 | \( 1 + 6.22T + 83T^{2} \) |
| 89 | \( 1 + 4.74T + 89T^{2} \) |
| 97 | \( 1 + 5.04T + 97T^{2} \) |
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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
−7.904387011286349671984957254510, −7.45009235220868819047093938540, −6.89592174540978437005166426180, −5.75066110639175680005303743730, −5.27548915948487723628126430947, −4.35158198556206738561240274824, −3.33976876639875764922556043736, −2.25355846588248715136308746074, −1.22301653701299104286698947725, 0,
1.22301653701299104286698947725, 2.25355846588248715136308746074, 3.33976876639875764922556043736, 4.35158198556206738561240274824, 5.27548915948487723628126430947, 5.75066110639175680005303743730, 6.89592174540978437005166426180, 7.45009235220868819047093938540, 7.904387011286349671984957254510
