| L(s) = 1 | + 5.26·2-s + 3·3-s + 19.6·4-s + 5·5-s + 15.7·6-s − 5.37·7-s + 61.4·8-s + 9·9-s + 26.3·10-s + 59.0·12-s + 5.15·13-s − 28.2·14-s + 15·15-s + 166.·16-s − 107.·17-s + 47.3·18-s + 70.1·19-s + 98.4·20-s − 16.1·21-s + 104.·23-s + 184.·24-s + 25·25-s + 27.1·26-s + 27·27-s − 105.·28-s + 122.·29-s + 78.9·30-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 0.577·3-s + 2.46·4-s + 0.447·5-s + 1.07·6-s − 0.290·7-s + 2.71·8-s + 0.333·9-s + 0.831·10-s + 1.42·12-s + 0.109·13-s − 0.539·14-s + 0.258·15-s + 2.59·16-s − 1.53·17-s + 0.620·18-s + 0.847·19-s + 1.10·20-s − 0.167·21-s + 0.947·23-s + 1.56·24-s + 0.200·25-s + 0.204·26-s + 0.192·27-s − 0.713·28-s + 0.784·29-s + 0.480·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.93746749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.93746749\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 5.26T + 8T^{2} \) |
| 7 | \( 1 + 5.37T + 343T^{2} \) |
| 13 | \( 1 - 5.15T + 2.19e3T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 70.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 122.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 459.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 310.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 322.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 425.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 75.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 898.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.19e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 277.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 156.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 96.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 840.T + 9.12e5T^{2} \) |
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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
−8.913756565237369572990137677964, −7.85444138206706893240258514378, −6.89946616569584366996095580564, −6.43991654416683018619593579753, −5.54557555296018949605193382116, −4.65373309934304937388015097981, −4.06578856020299078175997031077, −2.87830680894750883760537362042, −2.55413511531942314166241788734, −1.25566548165062588526259291134,
1.25566548165062588526259291134, 2.55413511531942314166241788734, 2.87830680894750883760537362042, 4.06578856020299078175997031077, 4.65373309934304937388015097981, 5.54557555296018949605193382116, 6.43991654416683018619593579753, 6.89946616569584366996095580564, 7.85444138206706893240258514378, 8.913756565237369572990137677964
