L(s) = 1 | − 8.89·5-s − 4.08·7-s + 4.37·11-s − 13.7i·13-s + 14.0·17-s + (−11.6 + 15.0i)19-s + 6.99·23-s + 54.0·25-s + 31.8i·29-s + 12.3i·31-s + 36.3·35-s − 4.89i·37-s + 73.1i·41-s + 23.8·43-s + 43.3·47-s + ⋯ |
L(s) = 1 | − 1.77·5-s − 0.584·7-s + 0.397·11-s − 1.05i·13-s + 0.828·17-s + (−0.611 + 0.791i)19-s + 0.304·23-s + 2.16·25-s + 1.09i·29-s + 0.398i·31-s + 1.03·35-s − 0.132i·37-s + 1.78i·41-s + 0.555·43-s + 0.922·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4144620870\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4144620870\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (11.6 - 15.0i)T \) |
good | 5 | \( 1 + 8.89T + 25T^{2} \) |
| 7 | \( 1 + 4.08T + 49T^{2} \) |
| 11 | \( 1 - 4.37T + 121T^{2} \) |
| 13 | \( 1 + 13.7iT - 169T^{2} \) |
| 17 | \( 1 - 14.0T + 289T^{2} \) |
| 23 | \( 1 - 6.99T + 529T^{2} \) |
| 29 | \( 1 - 31.8iT - 841T^{2} \) |
| 31 | \( 1 - 12.3iT - 961T^{2} \) |
| 37 | \( 1 + 4.89iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 73.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 23.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 43.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 65.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 62.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 70.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 126. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 44.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 0.118T + 5.32e3T^{2} \) |
| 79 | \( 1 - 19.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 97.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 15.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 6.98iT - 9.40e3T^{2} \) |
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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
−8.191326787023427699087635987400, −7.75743872664940689318590742428, −6.99511531650477091320824969295, −6.17456564421030967539696125933, −5.16707139203389204065114987948, −4.27734610895708507411791884675, −3.44806875082018580942187563650, −3.01854979613036296593553754732, −1.22169233767170067078461816722, −0.14098781794414136270204552200,
0.854879300747292382723430158308, 2.42802692904394001909069726566, 3.50171067855387353077154369875, 4.08146589805906329327751857650, 4.73648811877569745448235345127, 5.97861685513754853203763881654, 6.83079506244739651079320964422, 7.38858171424987906250310086253, 8.080936863175715557246065302776, 8.939872785445085207669278303401