L(s) = 1 | + (1.72 − 0.133i)3-s − 4.22·5-s + (2.37 − 1.15i)7-s + (2.96 − 0.460i)9-s − 1.92·11-s + (−0.291 + 0.504i)13-s + (−7.29 + 0.562i)15-s + (3.61 − 6.25i)17-s + (−2.10 − 3.64i)19-s + (3.95 − 2.31i)21-s − 1.27·23-s + 12.8·25-s + (5.05 − 1.18i)27-s + (−4.20 − 7.27i)29-s + (−0.476 − 0.824i)31-s + ⋯ |
L(s) = 1 | + (0.997 − 0.0769i)3-s − 1.88·5-s + (0.898 − 0.438i)7-s + (0.988 − 0.153i)9-s − 0.581·11-s + (−0.0808 + 0.140i)13-s + (−1.88 + 0.145i)15-s + (0.875 − 1.51i)17-s + (−0.482 − 0.835i)19-s + (0.862 − 0.506i)21-s − 0.266·23-s + 2.56·25-s + (0.973 − 0.228i)27-s + (−0.780 − 1.35i)29-s + (−0.0855 − 0.148i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.549345356\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.549345356\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.72 + 0.133i)T \) |
| 7 | \( 1 + (-2.37 + 1.15i)T \) |
good | 5 | \( 1 + 4.22T + 5T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 + (0.291 - 0.504i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.61 + 6.25i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.10 + 3.64i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 1.27T + 23T^{2} \) |
| 29 | \( 1 + (4.20 + 7.27i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.476 + 0.824i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.03 - 5.25i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.31 + 2.27i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.442 + 0.766i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.88 + 4.99i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.962 - 1.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.27 + 3.94i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.29 + 9.16i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.43 + 4.21i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + (-0.446 + 0.772i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.93 - 10.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.24 - 9.08i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.87 - 6.71i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.98 + 3.44i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.666362207143592102694308930613, −8.706214554861864399984397862201, −7.88197000686803735369166522813, −7.64496172115436084309528535070, −6.89232810530766841707686584090, −5.02087132071057525070059448328, −4.34040941612806404120105030543, −3.51445141611494966149299330619, −2.46802466886841608363819003924, −0.66805420763383641781291574207,
1.57167951023214223261118066732, 3.02421321870581285726587370387, 3.88353892028070845517458888555, 4.54458836533271786958605148962, 5.76556770250772133790784391992, 7.35968867849600876190475157390, 7.74423794216638466015357307808, 8.385071623493811879421329022501, 8.900873905101125929884483673710, 10.33861024412767054321709232493