L(s) = 1 | − 2·2-s + 4·4-s − 15.9·5-s + 7·7-s − 8·8-s + 31.8·10-s − 2.05·11-s + 79.6·13-s − 14·14-s + 16·16-s + 67.5·17-s − 108.·19-s − 63.7·20-s + 4.10·22-s + 7.74·23-s + 129.·25-s − 159.·26-s + 28·28-s − 249.·29-s + 5.30·31-s − 32·32-s − 135.·34-s − 111.·35-s − 48.6·37-s + 216.·38-s + 127.·40-s + 45.0·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.42·5-s + 0.377·7-s − 0.353·8-s + 1.00·10-s − 0.0561·11-s + 1.70·13-s − 0.267·14-s + 0.250·16-s + 0.964·17-s − 1.30·19-s − 0.713·20-s + 0.0397·22-s + 0.0702·23-s + 1.03·25-s − 1.20·26-s + 0.188·28-s − 1.59·29-s + 0.0307·31-s − 0.176·32-s − 0.681·34-s − 0.539·35-s − 0.216·37-s + 0.925·38-s + 0.504·40-s + 0.171·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
good | 5 | \( 1 + 15.9T + 125T^{2} \) |
| 11 | \( 1 + 2.05T + 1.33e3T^{2} \) |
| 13 | \( 1 - 79.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 7.74T + 1.21e4T^{2} \) |
| 29 | \( 1 + 249.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 5.30T + 2.97e4T^{2} \) |
| 37 | \( 1 + 48.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 45.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 201.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 4.49T + 1.03e5T^{2} \) |
| 53 | \( 1 + 279.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 675.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 842.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 329.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 811.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 635.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 480.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.67e3T + 9.12e5T^{2} \) |
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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
−10.87548278201405929715018866684, −9.419328707771239713712968757226, −8.339762220389741841391555892341, −8.007317160303903717525172164971, −6.92121125663947098563524768211, −5.77360164986486181078762450499, −4.21785657137222293266457649852, −3.33710530503998290147402861982, −1.47217450382155596059329704638, 0,
1.47217450382155596059329704638, 3.33710530503998290147402861982, 4.21785657137222293266457649852, 5.77360164986486181078762450499, 6.92121125663947098563524768211, 8.007317160303903717525172164971, 8.339762220389741841391555892341, 9.419328707771239713712968757226, 10.87548278201405929715018866684