L(s) = 1 | + (0.637 + 1.26i)2-s + (−1.18 + 1.61i)4-s + 3.42·5-s + 0.505i·7-s + (−2.78 − 0.469i)8-s + (2.18 + 4.32i)10-s − 3.31i·11-s + 5.43i·13-s + (−0.637 + 0.322i)14-s + (−1.18 − 3.82i)16-s − 1.58i·17-s − 6.74·19-s + (−4.06 + 5.51i)20-s + (4.18 − 2.11i)22-s + 4.30·23-s + ⋯ |
L(s) = 1 | + (0.451 + 0.892i)2-s + (−0.593 + 0.805i)4-s + 1.53·5-s + 0.191i·7-s + (−0.986 − 0.166i)8-s + (0.691 + 1.36i)10-s − 1.00i·11-s + 1.50i·13-s + (−0.170 + 0.0861i)14-s + (−0.296 − 0.955i)16-s − 0.384i·17-s − 1.54·19-s + (−0.908 + 1.23i)20-s + (0.892 − 0.451i)22-s + 0.897·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28298 + 1.08492i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28298 + 1.08492i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.637 - 1.26i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3.42T + 5T^{2} \) |
| 7 | \( 1 - 0.505iT - 7T^{2} \) |
| 11 | \( 1 + 3.31iT - 11T^{2} \) |
| 13 | \( 1 - 5.43iT - 13T^{2} \) |
| 17 | \( 1 + 1.58iT - 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 - 4.30T + 23T^{2} \) |
| 29 | \( 1 + 2.55T + 29T^{2} \) |
| 31 | \( 1 + 4.92iT - 31T^{2} \) |
| 37 | \( 1 + 7.45iT - 37T^{2} \) |
| 41 | \( 1 + 8.51iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 5.10T + 47T^{2} \) |
| 53 | \( 1 + 0.875T + 53T^{2} \) |
| 59 | \( 1 - 6.63iT - 59T^{2} \) |
| 61 | \( 1 - 10.8iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 9.40T + 71T^{2} \) |
| 73 | \( 1 + 3.74T + 73T^{2} \) |
| 79 | \( 1 - 6.44iT - 79T^{2} \) |
| 83 | \( 1 + 10.2iT - 83T^{2} \) |
| 89 | \( 1 + 3.75iT - 89T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−12.96920347537960604743801951418, −11.74172864154345246498941715674, −10.52831879173803804876308295172, −9.063570898159216293278642163148, −8.919686391248654679338182873616, −7.17369544057194018534886649291, −6.21296835242234956019807558402, −5.53452491685405563692100455047, −4.16336413846382076806159682866, −2.37137446776201990643666290341,
1.67387876487883170690795925035, 2.92953664176158627376357050248, 4.65202205570239063286729143235, 5.62994459270790815644211709391, 6.64251357576350219775876209725, 8.439207320806334264955447242970, 9.582790703848586812126255059710, 10.27290571112059143507105095248, 10.87696194498750572020822484258, 12.45107684291295739227361017555