L(s) = 1 | + (−1.73 − 0.0795i)3-s + (−0.139 + 2.23i)5-s + (0.622 + 2.32i)7-s + (2.98 + 0.275i)9-s + (−0.991 − 0.572i)11-s + (−0.640 + 2.38i)13-s + (0.419 − 3.85i)15-s + (−4.99 − 4.99i)17-s + 2.78i·19-s + (−0.891 − 4.06i)21-s + (5.95 + 1.59i)23-s + (−4.96 − 0.624i)25-s + (−5.14 − 0.713i)27-s + (0.672 − 1.16i)29-s + (−1.25 − 2.16i)31-s + ⋯ |
L(s) = 1 | + (−0.998 − 0.0459i)3-s + (−0.0625 + 0.998i)5-s + (0.235 + 0.877i)7-s + (0.995 + 0.0917i)9-s + (−0.299 − 0.172i)11-s + (−0.177 + 0.662i)13-s + (0.108 − 0.994i)15-s + (−1.21 − 1.21i)17-s + 0.638i·19-s + (−0.194 − 0.887i)21-s + (1.24 + 0.332i)23-s + (−0.992 − 0.124i)25-s + (−0.990 − 0.137i)27-s + (0.124 − 0.216i)29-s + (−0.224 − 0.389i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0996386 + 0.549116i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0996386 + 0.549116i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.73 + 0.0795i)T \) |
| 5 | \( 1 + (0.139 - 2.23i)T \) |
good | 7 | \( 1 + (-0.622 - 2.32i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.991 + 0.572i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.640 - 2.38i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (4.99 + 4.99i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.78iT - 19T^{2} \) |
| 23 | \( 1 + (-5.95 - 1.59i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.672 + 1.16i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.25 + 2.16i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.16 - 8.16i)T - 37iT^{2} \) |
| 41 | \( 1 + (1.70 - 0.986i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (8.68 - 2.32i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (11.9 - 3.19i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.84 + 1.84i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.31 + 2.27i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.54 - 6.13i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.0545 + 0.0146i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 9.10iT - 71T^{2} \) |
| 73 | \( 1 + (-7.82 - 7.82i)T + 73iT^{2} \) |
| 79 | \( 1 + (-8.46 - 4.88i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.724 + 2.70i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 4.87T + 89T^{2} \) |
| 97 | \( 1 + (-2.08 - 7.79i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−11.01198053684310512336231573433, −10.04320952748467829604680085125, −9.258857733027411922575716377133, −8.106781781562271455820961362988, −6.94016682878350236515047651363, −6.55660386856484967855769963647, −5.40566576783315558102441394532, −4.65969369327878622762233277498, −3.17482366978870731534640798191, −1.93674959580598665036654021708,
0.32409307073849917040205739430, 1.68463480651834838870166075577, 3.74747065412678968254112599235, 4.76141165783610066984201430197, 5.25784463399249565779784603974, 6.51983046745654827417674947311, 7.27840727525955196266479555236, 8.333740881049168706656173157313, 9.170608444095356774120855478128, 10.35697283715873145552256646480