| L(s) = 1 | + 1.70·2-s − 5.10·4-s − 5·5-s − 22.2·8-s − 8.50·10-s − 37.4·11-s − 29.0·13-s + 2.89·16-s + 58.4·17-s + 54.5·19-s + 25.5·20-s − 63.6·22-s − 161.·23-s + 25·25-s − 49.3·26-s − 137.·29-s − 154.·31-s + 183.·32-s + 99.4·34-s − 350.·37-s + 92.8·38-s + 111.·40-s + 353.·41-s − 518.·43-s + 190.·44-s − 275.·46-s − 542.·47-s + ⋯ |
| L(s) = 1 | + 0.601·2-s − 0.638·4-s − 0.447·5-s − 0.985·8-s − 0.269·10-s − 1.02·11-s − 0.619·13-s + 0.0452·16-s + 0.833·17-s + 0.659·19-s + 0.285·20-s − 0.616·22-s − 1.46·23-s + 0.200·25-s − 0.372·26-s − 0.880·29-s − 0.896·31-s + 1.01·32-s + 0.501·34-s − 1.55·37-s + 0.396·38-s + 0.440·40-s + 1.34·41-s − 1.83·43-s + 0.654·44-s − 0.881·46-s − 1.68·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8193963960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8193963960\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.70T + 8T^{2} \) |
| 11 | \( 1 + 37.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 29.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 350.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 518.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 542.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 305.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 14.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 171.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 551.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 120.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 284.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 941.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 377.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 677.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−8.552390438446042517550388443818, −7.928121915751284233212196480082, −7.28162703920366389412844994007, −6.10789216281293128148667186853, −5.29085316505930329359099282207, −4.85458081491807854820364249910, −3.71303461658186779984528906303, −3.21303841623059107067077121610, −1.94340258328469866153911550303, −0.35838979906258974904439614630,
0.35838979906258974904439614630, 1.94340258328469866153911550303, 3.21303841623059107067077121610, 3.71303461658186779984528906303, 4.85458081491807854820364249910, 5.29085316505930329359099282207, 6.10789216281293128148667186853, 7.28162703920366389412844994007, 7.928121915751284233212196480082, 8.552390438446042517550388443818
