| L(s) = 1 | + 9·3-s − 80.1·5-s + 221.·7-s + 81·9-s − 6.83·11-s + 1.03e3·13-s − 721.·15-s + 89.8·17-s − 361·19-s + 1.98e3·21-s − 1.93e3·23-s + 3.29e3·25-s + 729·27-s + 8.95e3·29-s + 152.·31-s − 61.5·33-s − 1.77e4·35-s − 2.97e3·37-s + 9.33e3·39-s − 1.38e4·41-s + 1.29e4·43-s − 6.49e3·45-s + 2.35e4·47-s + 3.20e4·49-s + 808.·51-s + 5.83e3·53-s + 547.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.43·5-s + 1.70·7-s + 0.333·9-s − 0.0170·11-s + 1.70·13-s − 0.827·15-s + 0.0753·17-s − 0.229·19-s + 0.984·21-s − 0.764·23-s + 1.05·25-s + 0.192·27-s + 1.97·29-s + 0.0285·31-s − 0.00983·33-s − 2.44·35-s − 0.356·37-s + 0.982·39-s − 1.28·41-s + 1.06·43-s − 0.477·45-s + 1.55·47-s + 1.90·49-s + 0.0435·51-s + 0.285·53-s + 0.0244·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.081429243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.081429243\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 19 | \( 1 + 361T \) |
| good | 5 | \( 1 + 80.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 221.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 6.83T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 89.8T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.93e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.95e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 152.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.97e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.38e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.83e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.03e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.63e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.35e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.62e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.51e4T + 8.58e9T^{2} \) |
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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
−8.897784268129144951316550507623, −8.353627742909534955952654615350, −7.962552581801861663621041012001, −7.10247662395789378208088052449, −5.87325470423380398872838337761, −4.55983362871247484343840447147, −4.12567291737108379475055083564, −3.10198549590267424408913053670, −1.72004815440458286339452448643, −0.811007455588589758684490061343,
0.811007455588589758684490061343, 1.72004815440458286339452448643, 3.10198549590267424408913053670, 4.12567291737108379475055083564, 4.55983362871247484343840447147, 5.87325470423380398872838337761, 7.10247662395789378208088052449, 7.962552581801861663621041012001, 8.353627742909534955952654615350, 8.897784268129144951316550507623
