L(s) = 1 | + (2.44 − 2.44i)5-s − 1.41i·7-s + (3.46 − 3.46i)11-s + (−1 − i)13-s + 4.89·17-s + (4.24 + 4.24i)19-s + 6.92i·23-s − 6.99i·25-s + (2.44 + 2.44i)29-s − 1.41·31-s + (−3.46 − 3.46i)35-s + (7 − 7i)37-s − 4.89i·41-s + (−4.24 + 4.24i)43-s − 6.92·47-s + ⋯ |
L(s) = 1 | + (1.09 − 1.09i)5-s − 0.534i·7-s + (1.04 − 1.04i)11-s + (−0.277 − 0.277i)13-s + 1.18·17-s + (0.973 + 0.973i)19-s + 1.44i·23-s − 1.39i·25-s + (0.454 + 0.454i)29-s − 0.254·31-s + (−0.585 − 0.585i)35-s + (1.15 − 1.15i)37-s − 0.765i·41-s + (−0.646 + 0.646i)43-s − 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.514259014\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.514259014\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.44 + 2.44i)T - 5iT^{2} \) |
| 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 + (-3.46 + 3.46i)T - 11iT^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + (-4.24 - 4.24i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.92iT - 23T^{2} \) |
| 29 | \( 1 + (-2.44 - 2.44i)T + 29iT^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + (-7 + 7i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.89iT - 41T^{2} \) |
| 43 | \( 1 + (4.24 - 4.24i)T - 43iT^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + (2.44 - 2.44i)T - 53iT^{2} \) |
| 59 | \( 1 + (6.92 - 6.92i)T - 59iT^{2} \) |
| 61 | \( 1 + (5 + 5i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5.65 - 5.65i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.8iT - 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + (-10.3 - 10.3i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.79iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.128757798541103631146528736544, −8.066990510012932111398128887740, −7.49362936295663227407650833190, −6.26946445336178959833555268548, −5.66876345632782040261208205527, −5.12061266887827726795817221233, −3.91747163867007589535351529227, −3.16635070409653421668687653711, −1.55016276352500378858196602241, −1.01330456036734425147560323510,
1.41265193476939621273408601493, 2.46460684292076548120397777403, 3.09613472086421004419943726993, 4.41657364598535375115895386963, 5.26028852004658240374846416466, 6.24355040894732990825513785778, 6.68973529506364756328895165344, 7.40135980712424972672807328673, 8.453564843979592734589419622563, 9.487046379169819664807538692188