L(s) = 1 | + 0.414·2-s − 7.82·4-s − 0.100·5-s − 6.55·8-s − 0.0416·10-s + 43.9·11-s + 16.6·13-s + 59.9·16-s − 121.·17-s + 127.·19-s + 0.786·20-s + 18.2·22-s − 53.5·23-s − 124.·25-s + 6.89·26-s − 235.·29-s + 18.7·31-s + 77.2·32-s − 50.3·34-s − 191.·37-s + 52.6·38-s + 0.658·40-s − 319.·41-s − 218.·43-s − 343.·44-s − 22.2·46-s + 401.·47-s + ⋯ |
L(s) = 1 | + 0.146·2-s − 0.978·4-s − 0.00898·5-s − 0.289·8-s − 0.00131·10-s + 1.20·11-s + 0.355·13-s + 0.936·16-s − 1.73·17-s + 1.53·19-s + 0.00879·20-s + 0.176·22-s − 0.485·23-s − 0.999·25-s + 0.0520·26-s − 1.50·29-s + 0.108·31-s + 0.426·32-s − 0.254·34-s − 0.852·37-s + 0.224·38-s + 0.00260·40-s − 1.21·41-s − 0.775·43-s − 1.17·44-s − 0.0711·46-s + 1.24·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 0.414T + 8T^{2} \) |
| 5 | \( 1 + 0.100T + 125T^{2} \) |
| 11 | \( 1 - 43.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 53.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 18.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 191.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 319.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 401.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 643.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 11.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 12.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 669.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 822.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 515.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 805.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 394.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 673.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.09e3T + 9.12e5T^{2} \) |
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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
−10.05657068243529147717160229682, −9.245276754124986470721385124011, −8.687597444370195160304228180508, −7.51279839970678956632749971633, −6.39273802010921929667185561391, −5.37504269197936174321461012309, −4.25862134111062709173977883295, −3.47535521764679561694509680839, −1.60954362499798667457814268582, 0,
1.60954362499798667457814268582, 3.47535521764679561694509680839, 4.25862134111062709173977883295, 5.37504269197936174321461012309, 6.39273802010921929667185561391, 7.51279839970678956632749971633, 8.687597444370195160304228180508, 9.245276754124986470721385124011, 10.05657068243529147717160229682