| L(s) = 1 | + 1.66·2-s − 5.22·4-s + 10.5·7-s − 22.0·8-s − 29.0·11-s + 23.5·13-s + 17.5·14-s + 5.19·16-s − 35.6·17-s + 23.7·19-s − 48.3·22-s + 78.9·23-s + 39.1·26-s − 55.2·28-s − 148.·29-s + 341.·31-s + 184.·32-s − 59.3·34-s + 337.·37-s + 39.5·38-s − 354.·41-s − 19.5·43-s + 152.·44-s + 131.·46-s + 112.·47-s − 231.·49-s − 122.·52-s + ⋯ |
| L(s) = 1 | + 0.588·2-s − 0.653·4-s + 0.570·7-s − 0.973·8-s − 0.797·11-s + 0.501·13-s + 0.335·14-s + 0.0811·16-s − 0.508·17-s + 0.286·19-s − 0.469·22-s + 0.715·23-s + 0.295·26-s − 0.373·28-s − 0.948·29-s + 1.98·31-s + 1.02·32-s − 0.299·34-s + 1.50·37-s + 0.168·38-s − 1.34·41-s − 0.0692·43-s + 0.521·44-s + 0.421·46-s + 0.348·47-s − 0.674·49-s − 0.327·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.66T + 8T^{2} \) |
| 7 | \( 1 - 10.5T + 343T^{2} \) |
| 11 | \( 1 + 29.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 23.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 78.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 341.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 354.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 19.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 112.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 699.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 27.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 825.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 825.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 337.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 717.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 310.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 552.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.16e3T + 9.12e5T^{2} \) |
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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
−8.277941551356578001635333287715, −7.85498061967060483779493876737, −6.63032285002668757427647061322, −5.86919285335838528023449405190, −4.95462265167959461849106432464, −4.53697189944617230630291749269, −3.45264748982714196380905522323, −2.60870054024066104567193722063, −1.22741049025787467493528134708, 0,
1.22741049025787467493528134708, 2.60870054024066104567193722063, 3.45264748982714196380905522323, 4.53697189944617230630291749269, 4.95462265167959461849106432464, 5.86919285335838528023449405190, 6.63032285002668757427647061322, 7.85498061967060483779493876737, 8.277941551356578001635333287715
