| L(s) = 1 | + 6.89·3-s + 12.6·7-s + 20.5·9-s − 59.1·11-s + 42.2·13-s − 126.·17-s + 19.1·19-s + 87.5·21-s − 78.3·23-s − 44.1·27-s + 148.·29-s − 139.·31-s − 408.·33-s − 66.5·37-s + 291.·39-s − 203.·41-s + 288.·43-s − 360.·47-s − 181.·49-s − 871.·51-s − 686.·53-s + 132.·57-s + 83.1·59-s + 208.·61-s + 261.·63-s + 192.·67-s − 540.·69-s + ⋯ |
| L(s) = 1 | + 1.32·3-s + 0.685·7-s + 0.762·9-s − 1.62·11-s + 0.900·13-s − 1.80·17-s + 0.231·19-s + 0.910·21-s − 0.709·23-s − 0.314·27-s + 0.950·29-s − 0.806·31-s − 2.15·33-s − 0.295·37-s + 1.19·39-s − 0.774·41-s + 1.02·43-s − 1.11·47-s − 0.529·49-s − 2.39·51-s − 1.78·53-s + 0.307·57-s + 0.183·59-s + 0.438·61-s + 0.522·63-s + 0.350·67-s − 0.942·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 6.89T + 27T^{2} \) |
| 7 | \( 1 - 12.6T + 343T^{2} \) |
| 11 | \( 1 + 59.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 19.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 78.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 66.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 203.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 288.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 360.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 686.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 83.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 208.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 192.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 500.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 122.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 289.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 573.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 565.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 643.T + 9.12e5T^{2} \) |
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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
−8.376340701054261650970341601002, −8.204106409983130412150326861610, −7.31838753956434349267269458068, −6.29053344439092067155925801829, −5.18359255466509856547032800004, −4.36098729910368239879717081954, −3.33395909779314677121704238564, −2.46016094269881458223066331061, −1.71916682392707735102433056662, 0,
1.71916682392707735102433056662, 2.46016094269881458223066331061, 3.33395909779314677121704238564, 4.36098729910368239879717081954, 5.18359255466509856547032800004, 6.29053344439092067155925801829, 7.31838753956434349267269458068, 8.204106409983130412150326861610, 8.376340701054261650970341601002
