L(s) = 1 | + 4.85·2-s + 15.5·4-s + 16.2·7-s + 36.5·8-s − 59.5·11-s + 27.0·13-s + 78.7·14-s + 53.0·16-s − 78.9·17-s − 142.·19-s − 288.·22-s − 85.6·23-s + 131.·26-s + 252.·28-s − 226.·29-s − 150.·31-s − 35.0·32-s − 382.·34-s + 234.·37-s − 690.·38-s − 73.8·41-s + 267.·43-s − 924.·44-s − 415.·46-s − 266.·47-s − 79.3·49-s + 420.·52-s + ⋯ |
L(s) = 1 | + 1.71·2-s + 1.94·4-s + 0.876·7-s + 1.61·8-s − 1.63·11-s + 0.577·13-s + 1.50·14-s + 0.829·16-s − 1.12·17-s − 1.71·19-s − 2.79·22-s − 0.776·23-s + 0.990·26-s + 1.70·28-s − 1.44·29-s − 0.873·31-s − 0.193·32-s − 1.93·34-s + 1.04·37-s − 2.94·38-s − 0.281·41-s + 0.947·43-s − 3.16·44-s − 1.33·46-s − 0.825·47-s − 0.231·49-s + 1.12·52-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 | \( 1 - 4.85T + 8T^{2} \) |
| 7 | \( 1 - 16.2T + 343T^{2} \) |
| 11 | \( 1 + 59.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 85.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 150.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 234.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 73.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 267.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 266.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 603.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 508.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 78.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 488.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 73.2T + 3.57e5T^{2} \) |
| 73 | \( 1 - 115.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 782.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 271.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 211.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.66e3T + 9.12e5T^{2} \) |
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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
−8.141843490324596630511408488858, −7.49815729032289622690229993449, −6.47972658223186164116026086338, −5.80361300755507250787413278101, −5.06523750313695219531025110687, −4.37726297754965478785202165392, −3.67455174563280715953167162220, −2.38657425812594927714958365594, −1.99088967485610600731475839076, 0,
1.99088967485610600731475839076, 2.38657425812594927714958365594, 3.67455174563280715953167162220, 4.37726297754965478785202165392, 5.06523750313695219531025110687, 5.80361300755507250787413278101, 6.47972658223186164116026086338, 7.49815729032289622690229993449, 8.141843490324596630511408488858