L(s) = 1 | + (1.65 − 0.522i)3-s + (0.707 + 0.707i)7-s + (2.45 − 1.72i)9-s + 3.21i·11-s + (−3.87 + 3.87i)13-s + (4.34 − 4.34i)17-s + 6.27i·19-s + (1.53 + 0.797i)21-s + (−2.64 − 2.64i)23-s + (3.14 − 4.13i)27-s + 8.25·29-s + 2.30·31-s + (1.67 + 5.30i)33-s + (5.79 + 5.79i)37-s + (−4.36 + 8.41i)39-s + ⋯ |
L(s) = 1 | + (0.953 − 0.301i)3-s + (0.267 + 0.267i)7-s + (0.817 − 0.575i)9-s + 0.968i·11-s + (−1.07 + 1.07i)13-s + (1.05 − 1.05i)17-s + 1.44i·19-s + (0.335 + 0.174i)21-s + (−0.550 − 0.550i)23-s + (0.605 − 0.795i)27-s + 1.53·29-s + 0.414·31-s + (0.292 + 0.922i)33-s + (0.952 + 0.952i)37-s + (−0.699 + 1.34i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.570379166\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.570379166\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.65 + 0.522i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 - 3.21iT - 11T^{2} \) |
| 13 | \( 1 + (3.87 - 3.87i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.34 + 4.34i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.27iT - 19T^{2} \) |
| 23 | \( 1 + (2.64 + 2.64i)T + 23iT^{2} \) |
| 29 | \( 1 - 8.25T + 29T^{2} \) |
| 31 | \( 1 - 2.30T + 31T^{2} \) |
| 37 | \( 1 + (-5.79 - 5.79i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.59iT - 41T^{2} \) |
| 43 | \( 1 + (3.88 - 3.88i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.57 + 1.57i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.48 + 2.48i)T + 53iT^{2} \) |
| 59 | \( 1 - 0.317T + 59T^{2} \) |
| 61 | \( 1 + 3.94T + 61T^{2} \) |
| 67 | \( 1 + (-4.21 - 4.21i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.52iT - 71T^{2} \) |
| 73 | \( 1 + (-1.46 + 1.46i)T - 73iT^{2} \) |
| 79 | \( 1 + 12.9iT - 79T^{2} \) |
| 83 | \( 1 + (-9.41 - 9.41i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.62T + 89T^{2} \) |
| 97 | \( 1 + (5.42 + 5.42i)T + 97iT^{2} \) |
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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
−9.307923718533209247567913399278, −8.122900245632237837859107065702, −7.86693489906784958919807221694, −6.92602690601335573728806123270, −6.26780031419420434242181586506, −4.85979005404541837941521750940, −4.40077556585818768403693846629, −3.13882143650497308419440173658, −2.32479981693808423731799124645, −1.37348708226364902271668263163,
0.873537005925984952100916035595, 2.36279534397924317747962543576, 3.13783766096170511420367302660, 3.98676820597683016488679506762, 4.97020954315725637999365944725, 5.72911111429665342811455233253, 6.89347115705239495070150599749, 7.76405469944471232142781405914, 8.179339599353835743527163917931, 8.964763859624892210846677829870