| L(s) = 1 | + 2.24·2-s − 1.84·3-s − 2.97·4-s − 18.3·5-s − 4.13·6-s − 24.6·8-s − 23.6·9-s − 41.0·10-s + 52.4·11-s + 5.48·12-s + 87.5·13-s + 33.7·15-s − 31.3·16-s − 15.4·17-s − 52.8·18-s − 67.0·19-s + 54.5·20-s + 117.·22-s + 132.·23-s + 45.3·24-s + 211.·25-s + 196.·26-s + 93.2·27-s − 29·29-s + 75.7·30-s − 90.2·31-s + 126.·32-s + ⋯ |
| L(s) = 1 | + 0.792·2-s − 0.354·3-s − 0.372·4-s − 1.63·5-s − 0.281·6-s − 1.08·8-s − 0.874·9-s − 1.29·10-s + 1.43·11-s + 0.131·12-s + 1.86·13-s + 0.581·15-s − 0.489·16-s − 0.219·17-s − 0.692·18-s − 0.809·19-s + 0.610·20-s + 1.13·22-s + 1.20·23-s + 0.385·24-s + 1.68·25-s + 1.48·26-s + 0.664·27-s − 0.185·29-s + 0.460·30-s − 0.522·31-s + 0.699·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 2.24T + 8T^{2} \) |
| 3 | \( 1 + 1.84T + 27T^{2} \) |
| 5 | \( 1 + 18.3T + 125T^{2} \) |
| 11 | \( 1 - 52.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 87.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 67.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 132.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 90.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 11.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 18.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 21.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 337.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 84.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 330.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 492.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 347.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 986.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 594.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 334.T + 9.12e5T^{2} \) |
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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
−8.653009190000174792566726043112, −8.228613230263023213578184873141, −6.84214982390588541727741246447, −6.28258206270891185606134297318, −5.32306825531168414029152127618, −4.27427196030925584701096633708, −3.79613537310271263041257020249, −3.11752676325919430580275549327, −1.06905004813252945644876899645, 0,
1.06905004813252945644876899645, 3.11752676325919430580275549327, 3.79613537310271263041257020249, 4.27427196030925584701096633708, 5.32306825531168414029152127618, 6.28258206270891185606134297318, 6.84214982390588541727741246447, 8.228613230263023213578184873141, 8.653009190000174792566726043112
