| L(s) = 1 | − 2-s + 3-s + 4-s + 0.0176·5-s − 6-s + 0.282·7-s − 8-s + 9-s − 0.0176·10-s − 2.15·11-s + 12-s − 0.142·13-s − 0.282·14-s + 0.0176·15-s + 16-s + 4.89·17-s − 18-s − 19-s + 0.0176·20-s + 0.282·21-s + 2.15·22-s + 1.73·23-s − 24-s − 4.99·25-s + 0.142·26-s + 27-s + 0.282·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.00787·5-s − 0.408·6-s + 0.106·7-s − 0.353·8-s + 0.333·9-s − 0.00557·10-s − 0.650·11-s + 0.288·12-s − 0.0394·13-s − 0.0754·14-s + 0.00454·15-s + 0.250·16-s + 1.18·17-s − 0.235·18-s − 0.229·19-s + 0.00393·20-s + 0.0616·21-s + 0.459·22-s + 0.362·23-s − 0.204·24-s − 0.999·25-s + 0.0279·26-s + 0.192·27-s + 0.0533·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 - 0.0176T + 5T^{2} \) |
| 7 | \( 1 - 0.282T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 + 0.142T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 23 | \( 1 - 1.73T + 23T^{2} \) |
| 29 | \( 1 + 6.91T + 29T^{2} \) |
| 31 | \( 1 + 0.562T + 31T^{2} \) |
| 37 | \( 1 + 3.30T + 37T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 - 1.91T + 43T^{2} \) |
| 47 | \( 1 + 5.14T + 47T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 5.18T + 73T^{2} \) |
| 79 | \( 1 - 6.71T + 79T^{2} \) |
| 83 | \( 1 - 3.07T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.76548977575190179722828627790, −7.36554027564742730546134954015, −6.43970893128472632589028103520, −5.61988019289138704540483690321, −4.92681000991338604720075550042, −3.77240132760549587013745530698, −3.15489577448068846709866876203, −2.18785774163980407936705757109, −1.40348860005392889045807273703, 0,
1.40348860005392889045807273703, 2.18785774163980407936705757109, 3.15489577448068846709866876203, 3.77240132760549587013745530698, 4.92681000991338604720075550042, 5.61988019289138704540483690321, 6.43970893128472632589028103520, 7.36554027564742730546134954015, 7.76548977575190179722828627790
