| L(s) = 1 | − 3-s − 4·5-s − 2·7-s + 9-s − 4·11-s + 13-s + 4·15-s + 2·17-s − 2·19-s + 2·21-s + 11·25-s − 27-s − 6·29-s − 10·31-s + 4·33-s + 8·35-s + 10·37-s − 39-s + 8·41-s + 4·43-s − 4·45-s − 4·47-s − 3·49-s − 2·51-s − 10·53-s + 16·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 1.03·15-s + 0.485·17-s − 0.458·19-s + 0.436·21-s + 11/5·25-s − 0.192·27-s − 1.11·29-s − 1.79·31-s + 0.696·33-s + 1.35·35-s + 1.64·37-s − 0.160·39-s + 1.24·41-s + 0.609·43-s − 0.596·45-s − 0.583·47-s − 3/7·49-s − 0.280·51-s − 1.37·53-s + 2.15·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.60735281309983639791609167415, −11.28731824529632737026100079618, −10.81735681549710646583055818250, −9.399166260213490339159845776420, −7.965645572263157998665195857010, −7.33786979345918290465240439010, −5.89249065254499898615542236081, −4.45984308910786813852795903758, −3.27214076659366118813611445645, 0,
3.27214076659366118813611445645, 4.45984308910786813852795903758, 5.89249065254499898615542236081, 7.33786979345918290465240439010, 7.965645572263157998665195857010, 9.399166260213490339159845776420, 10.81735681549710646583055818250, 11.28731824529632737026100079618, 12.60735281309983639791609167415
