L(s) = 1 | + (−1.62 + 0.594i)3-s + (−0.0563 − 0.0563i)5-s − 7-s + (2.29 − 1.93i)9-s + (0.982 − 0.982i)11-s + (−0.649 − 0.649i)13-s + (0.125 + 0.0581i)15-s + 4.55i·17-s + (−2.11 + 2.11i)19-s + (1.62 − 0.594i)21-s − 2.24i·23-s − 4.99i·25-s + (−2.57 + 4.51i)27-s + (−4.36 + 4.36i)29-s − 4.82i·31-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.343i)3-s + (−0.0251 − 0.0251i)5-s − 0.377·7-s + (0.764 − 0.645i)9-s + (0.296 − 0.296i)11-s + (−0.180 − 0.180i)13-s + (0.0323 + 0.0150i)15-s + 1.10i·17-s + (−0.484 + 0.484i)19-s + (0.354 − 0.129i)21-s − 0.467i·23-s − 0.998i·25-s + (−0.496 + 0.868i)27-s + (−0.810 + 0.810i)29-s − 0.867i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4404655292\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4404655292\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.62 - 0.594i)T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (0.0563 + 0.0563i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.982 + 0.982i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.649 + 0.649i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.55iT - 17T^{2} \) |
| 19 | \( 1 + (2.11 - 2.11i)T - 19iT^{2} \) |
| 23 | \( 1 + 2.24iT - 23T^{2} \) |
| 29 | \( 1 + (4.36 - 4.36i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.82iT - 31T^{2} \) |
| 37 | \( 1 + (1.58 - 1.58i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.28T + 41T^{2} \) |
| 43 | \( 1 + (7.65 + 7.65i)T + 43iT^{2} \) |
| 47 | \( 1 - 1.18T + 47T^{2} \) |
| 53 | \( 1 + (9.36 + 9.36i)T + 53iT^{2} \) |
| 59 | \( 1 + (3.21 - 3.21i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.0261 - 0.0261i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7.39 + 7.39i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.08iT - 71T^{2} \) |
| 73 | \( 1 + 14.0iT - 73T^{2} \) |
| 79 | \( 1 + 14.7iT - 79T^{2} \) |
| 83 | \( 1 + (2.06 + 2.06i)T + 83iT^{2} \) |
| 89 | \( 1 - 6.33T + 89T^{2} \) |
| 97 | \( 1 + 7.26T + 97T^{2} \) |
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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
−9.487841854500972751539346428872, −8.618326478381723210563355111094, −7.69655587061965214836714706278, −6.55293365609285559658043417713, −6.14514450968705601230145605806, −5.19471735719700541992599343780, −4.20754335461926807937170901559, −3.43339602627887944566356345578, −1.81137513025053304277488944814, −0.21468614987547240385668485048,
1.33529752614527921188716768259, 2.66347399685633935528466282763, 4.01124836918361606971076842260, 4.95795517612346299764533632418, 5.71103862200108258482598516541, 6.72540379327085181052756298782, 7.16932007498439780737008140553, 8.092472060224821277248444076563, 9.396068365688103163736829794105, 9.703498648078352309443698003465