| L(s) = 1 | + 4.86·2-s + 15.7·4-s + 5·5-s + 37.5·8-s + 24.3·10-s + 8.38·11-s + 92.3·13-s + 57.0·16-s − 48.7·17-s − 50.3·19-s + 78.5·20-s + 40.8·22-s + 55.5·23-s + 25·25-s + 449.·26-s − 159.·29-s + 232.·31-s − 22.5·32-s − 237.·34-s + 297.·37-s − 245.·38-s + 187.·40-s + 433.·41-s + 59.2·43-s + 131.·44-s + 270.·46-s + 349.·47-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 1.96·4-s + 0.447·5-s + 1.65·8-s + 0.769·10-s + 0.229·11-s + 1.96·13-s + 0.890·16-s − 0.695·17-s − 0.607·19-s + 0.877·20-s + 0.395·22-s + 0.503·23-s + 0.200·25-s + 3.38·26-s − 1.02·29-s + 1.34·31-s − 0.124·32-s − 1.19·34-s + 1.32·37-s − 1.04·38-s + 0.741·40-s + 1.65·41-s + 0.210·43-s + 0.451·44-s + 0.867·46-s + 1.08·47-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
| \(L(2)\) |
\(\approx\) |
\(8.549044955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.549044955\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.86T + 8T^{2} \) |
| 11 | \( 1 - 8.38T + 1.33e3T^{2} \) |
| 13 | \( 1 - 92.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 55.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 159.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 232.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 297.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 433.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 59.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 349.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 267.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 515.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 664.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 556.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 226.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 337.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 305.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 323.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 542.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.42e3T + 9.12e5T^{2} \) |
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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
−8.715397229872806696139142948090, −7.69451070891967816829407269550, −6.63743519316301348998235885037, −6.14516337079704734338833771047, −5.63790122043245830282839929581, −4.48153866913418107390363783403, −4.02927327502931147304371457866, −3.06049938525845899702651765654, −2.18665785386605808300940694869, −1.08043797700520971062940703726,
1.08043797700520971062940703726, 2.18665785386605808300940694869, 3.06049938525845899702651765654, 4.02927327502931147304371457866, 4.48153866913418107390363783403, 5.63790122043245830282839929581, 6.14516337079704734338833771047, 6.63743519316301348998235885037, 7.69451070891967816829407269550, 8.715397229872806696139142948090
