L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−1.52 + 1.10i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + 1.87·10-s + (−2.97 + 1.46i)11-s + 12-s + (−0.748 − 0.543i)13-s + (0.309 − 0.951i)14-s + (0.580 + 1.78i)15-s + (−0.809 + 0.587i)16-s + (−3.66 + 2.66i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 + 0.475i)4-s + (−0.679 + 0.493i)5-s + (−0.330 + 0.239i)6-s + (0.116 + 0.359i)7-s + (0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + 0.594·10-s + (−0.896 + 0.442i)11-s + 0.288·12-s + (−0.207 − 0.150i)13-s + (0.0825 − 0.254i)14-s + (0.149 + 0.461i)15-s + (−0.202 + 0.146i)16-s + (−0.888 + 0.645i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.269929 + 0.319587i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.269929 + 0.319587i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (2.97 - 1.46i)T \) |
good | 5 | \( 1 + (1.52 - 1.10i)T + (1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (0.748 + 0.543i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (3.66 - 2.66i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.858 - 2.64i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 4.11T + 23T^{2} \) |
| 29 | \( 1 + (-1.44 - 4.43i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.635 + 0.461i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.344 - 1.06i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.146 + 0.451i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + (0.574 - 1.76i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.70 + 2.69i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.458 + 1.41i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (11.5 - 8.40i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 0.676T + 67T^{2} \) |
| 71 | \( 1 + (-4.30 + 3.12i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.08 + 3.33i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-3.23 - 2.34i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (2.55 - 1.85i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + (3.82 + 2.78i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.14308024412223177261777062037, −10.57902071583149365021372534958, −9.506447371058650106674450058102, −8.432355469233964293782123958232, −7.79320558124005265760422402886, −7.00994891604746872521239023204, −5.82740892216714078994617342594, −4.29364122228638603756250219864, −3.01832185698129161496242803680, −1.92016952337529232773148451986,
0.28761356283273995710718912295, 2.53257925801006396668733239591, 4.14852514782513353695694717773, 4.94136482016403636377809237446, 6.16766909239340280596010048917, 7.43034350383457085598826473988, 8.118522031374734007940552154437, 8.919703850028049758995187561567, 9.788305549776850400763293740234, 10.75276593586976357923691392728