L(s) = 1 | + (0.309 + 0.951i)3-s + 1.61·7-s + (0.951 + 1.30i)13-s + (−0.587 − 0.190i)19-s + (0.500 + 1.53i)21-s + (−0.809 − 0.587i)23-s + (0.809 + 0.587i)27-s + (0.190 + 0.587i)29-s + (−0.951 − 0.309i)31-s + (−0.587 − 0.809i)37-s + (−0.951 + 1.30i)39-s + 0.618·43-s + (−0.5 − 1.53i)47-s + 1.61·49-s + (0.951 − 0.309i)53-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)3-s + 1.61·7-s + (0.951 + 1.30i)13-s + (−0.587 − 0.190i)19-s + (0.500 + 1.53i)21-s + (−0.809 − 0.587i)23-s + (0.809 + 0.587i)27-s + (0.190 + 0.587i)29-s + (−0.951 − 0.309i)31-s + (−0.587 − 0.809i)37-s + (−0.951 + 1.30i)39-s + 0.618·43-s + (−0.5 − 1.53i)47-s + 1.61·49-s + (0.951 − 0.309i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.852113061\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.852113061\) |
\(L(1)\) |
|
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 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 - 1.61T + T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
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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
−8.808931071875371729101214178564, −8.283815195636341383433668628851, −7.33452253350922812040502561672, −6.59757824707665936752858036506, −5.59368679924210109415420085198, −4.80495157789258978762816413539, −4.15129321246153231689223751688, −3.70951708029968022373038195566, −2.24978459082322600216950942499, −1.49102229140475221554055949740,
1.21359031867850307218632711946, 1.80453810769698457888590891357, 2.81318755090589922217851250374, 3.94000938596300812377468758890, 4.76593201985756159835433121293, 5.63872156075381401222679002498, 6.26041191177197082492048374968, 7.38465477145024823501429647181, 7.76266676546745900336736965414, 8.337278805969994546519825837253