L(s) = 1 | + (0.258 − 0.965i)2-s + (−1.62 − 0.599i)3-s + (−0.866 − 0.499i)4-s + (−1 + 1.41i)6-s + (4.40 + 1.18i)7-s + (−0.707 + 0.707i)8-s + (2.28 + 1.94i)9-s + (−0.550 + 0.317i)11-s + (1.10 + 1.33i)12-s + (3.34 − 0.896i)13-s + (2.28 − 3.94i)14-s + (0.500 + 0.866i)16-s + (−0.317 − 0.317i)17-s + (2.47 − 1.69i)18-s − 6.44i·19-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.938 − 0.346i)3-s + (−0.433 − 0.249i)4-s + (−0.408 + 0.577i)6-s + (1.66 + 0.446i)7-s + (−0.249 + 0.249i)8-s + (0.760 + 0.649i)9-s + (−0.165 + 0.0958i)11-s + (0.319 + 0.384i)12-s + (0.928 − 0.248i)13-s + (0.609 − 1.05i)14-s + (0.125 + 0.216i)16-s + (−0.0770 − 0.0770i)17-s + (0.582 − 0.400i)18-s − 1.47i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05607 - 0.793088i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05607 - 0.793088i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 + (1.62 + 0.599i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-4.40 - 1.18i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.550 - 0.317i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.34 + 0.896i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (0.317 + 0.317i)T + 17iT^{2} \) |
| 19 | \( 1 + 6.44iT - 19T^{2} \) |
| 23 | \( 1 + (-0.258 - 0.965i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.158 + 0.275i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.224 - 0.389i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 + 3i)T - 37iT^{2} \) |
| 41 | \( 1 + (-6.39 - 3.69i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.896 + 3.34i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-2.32 + 8.69i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (3.78 - 3.78i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.48 - 7.77i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.275 - 0.476i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.71 - 6.38i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 6.29iT - 71T^{2} \) |
| 73 | \( 1 + (-6.89 - 6.89i)T + 73iT^{2} \) |
| 79 | \( 1 + (2.12 - 1.22i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.26 + 1.41i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 8.02T + 89T^{2} \) |
| 97 | \( 1 + (-2.59 - 0.695i)T + (84.0 + 48.5i)T^{2} \) |
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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
−11.19452317308033456401532889539, −10.47800913263924880700264345631, −9.135223595404636406784507557362, −8.219280096124776323083175024795, −7.22702974032688983233424859040, −5.88391465346115219431326956751, −5.09613220576042151324586246553, −4.27056287314976509738925932514, −2.38147510110409993279539022678, −1.13513997621708988247559467526,
1.37523153071072145077868778629, 3.87718060518937104688325882736, 4.64811039037155288436586154338, 5.59714349726953349854864446512, 6.42014317926077110222500434002, 7.65045590647401785810565236041, 8.272068635404070655448173930867, 9.477188440132685772999814489977, 10.65489077642635586558703071637, 11.14410339422671742857607329228