| L(s) = 1 | − 6.22·3-s − 9.40·5-s − 204.·9-s + 765.·11-s − 732.·13-s + 58.5·15-s − 482.·17-s + 2.61e3·19-s − 535.·23-s − 3.03e3·25-s + 2.78e3·27-s + 3.94e3·29-s − 3.66e3·31-s − 4.76e3·33-s + 1.26e4·37-s + 4.55e3·39-s − 4.81e3·41-s − 4.93e3·43-s + 1.92e3·45-s − 1.74e4·47-s + 3.00e3·51-s − 4.65e3·53-s − 7.19e3·55-s − 1.62e4·57-s − 3.25e4·59-s + 4.34e4·61-s + 6.88e3·65-s + ⋯ |
| L(s) = 1 | − 0.399·3-s − 0.168·5-s − 0.840·9-s + 1.90·11-s − 1.20·13-s + 0.0671·15-s − 0.404·17-s + 1.66·19-s − 0.211·23-s − 0.971·25-s + 0.735·27-s + 0.870·29-s − 0.684·31-s − 0.761·33-s + 1.51·37-s + 0.479·39-s − 0.447·41-s − 0.407·43-s + 0.141·45-s − 1.14·47-s + 0.161·51-s − 0.227·53-s − 0.320·55-s − 0.664·57-s − 1.21·59-s + 1.49·61-s + 0.202·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 6.22T + 243T^{2} \) |
| 5 | \( 1 + 9.40T + 3.12e3T^{2} \) |
| 11 | \( 1 - 765.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 732.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 482.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.61e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 535.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.66e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.81e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.93e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.74e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.34e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.54e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.94e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.08e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.45e4T + 8.58e9T^{2} \) |
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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
−9.832360189877354889709179011601, −9.303396218052931818621957089435, −8.171861439130746379331793224359, −7.09833167037729471603305243618, −6.22229739107787127252586273799, −5.19413423380865450157836079803, −4.07599755701363741036760484331, −2.87023363221993727854004884640, −1.34118169975096379317879668229, 0,
1.34118169975096379317879668229, 2.87023363221993727854004884640, 4.07599755701363741036760484331, 5.19413423380865450157836079803, 6.22229739107787127252586273799, 7.09833167037729471603305243618, 8.171861439130746379331793224359, 9.303396218052931818621957089435, 9.832360189877354889709179011601
