L(s) = 1 | + (−0.587 + 0.809i)3-s − 0.618i·7-s + (−0.190 − 0.587i)13-s + (−0.951 − 1.30i)19-s + (0.500 + 0.363i)21-s + (−0.951 − 0.309i)23-s + (−0.951 − 0.309i)27-s + (−1.30 − 0.951i)29-s + (−0.587 − 0.809i)31-s + (0.309 + 0.951i)37-s + (0.587 + 0.190i)39-s + 1.61i·43-s + (0.363 − 0.5i)47-s + 0.618·49-s + (−0.809 − 0.587i)53-s + ⋯ |
L(s) = 1 | + (−0.587 + 0.809i)3-s − 0.618i·7-s + (−0.190 − 0.587i)13-s + (−0.951 − 1.30i)19-s + (0.500 + 0.363i)21-s + (−0.951 − 0.309i)23-s + (−0.951 − 0.309i)27-s + (−1.30 − 0.951i)29-s + (−0.587 − 0.809i)31-s + (0.309 + 0.951i)37-s + (0.587 + 0.190i)39-s + 1.61i·43-s + (0.363 − 0.5i)47-s + 0.618·49-s + (−0.809 − 0.587i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4221169199\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4221169199\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + 0.618iT - T^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - 1.61iT - T^{2} \) |
| 47 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
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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
−8.313499975448239899849433853385, −7.74935139754921637088016768507, −6.92790392731338462199450633641, −6.04986637202879524979403857795, −5.43245289164208157077884388030, −4.38175614992658115751493001002, −4.24646235589071134660338899228, −2.99843782986114989786713229061, −1.93401610466446776177180549580, −0.24087963768786083539431845952,
1.55341484233124593480982043600, 2.13623862045543023554874664814, 3.51656931982201179403217121866, 4.22738752948814305117997423685, 5.48253445940003843802495563250, 5.84647250114383527713010531185, 6.60219322402632972673554619464, 7.33237182223157340463789738687, 7.919817914668191437967980547287, 8.963973019650543863098148949657