| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 4·11-s + 12-s − 6·13-s − 14-s + 16-s − 2·17-s − 18-s − 4·19-s + 21-s + 4·22-s − 8·23-s − 24-s + 6·26-s + 27-s + 28-s − 2·29-s − 32-s − 4·33-s + 2·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.218·21-s + 0.852·22-s − 1.66·23-s − 0.204·24-s + 1.17·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.176·32-s − 0.696·33-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−19.70988803782865, −19.07359182267525, −18.33458532905965, −17.76516030979300, −17.06998648859395, −16.27116890492155, −15.54549458952509, −14.86774343593110, −14.31643658597677, −13.28720768503600, −12.61086972138551, −11.81224716102558, −10.88037870677058, −10.10222911236605, −9.609539393852220, −8.562619095071442, −7.885745778855407, −7.380325109193969, −6.263292807161310, −5.123783494769916, −4.166958040114005, −2.646084597318552, −2.062226348106009, 0,
2.062226348106009, 2.646084597318552, 4.166958040114005, 5.123783494769916, 6.263292807161310, 7.380325109193969, 7.885745778855407, 8.562619095071442, 9.609539393852220, 10.10222911236605, 10.88037870677058, 11.81224716102558, 12.61086972138551, 13.28720768503600, 14.31643658597677, 14.86774343593110, 15.54549458952509, 16.27116890492155, 17.06998648859395, 17.76516030979300, 18.33458532905965, 19.07359182267525, 19.70988803782865
