L(s) = 1 | + 2-s + (−0.707 − 0.707i)3-s + 4-s + (0.906 + 2.04i)5-s + (−0.707 − 0.707i)6-s + (1.03 + 1.03i)7-s + 8-s + 1.00i·9-s + (0.906 + 2.04i)10-s − 6.26i·11-s + (−0.707 − 0.707i)12-s + 0.736·13-s + (1.03 + 1.03i)14-s + (0.804 − 2.08i)15-s + 16-s + 2.79i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (0.405 + 0.914i)5-s + (−0.288 − 0.288i)6-s + (0.390 + 0.390i)7-s + 0.353·8-s + 0.333i·9-s + (0.286 + 0.646i)10-s − 1.88i·11-s + (−0.204 − 0.204i)12-s + 0.204·13-s + (0.276 + 0.276i)14-s + (0.207 − 0.538i)15-s + 0.250·16-s + 0.677i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.566760058\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.566760058\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.906 - 2.04i)T \) |
| 37 | \( 1 + (-4.27 - 4.32i)T \) |
good | 7 | \( 1 + (-1.03 - 1.03i)T + 7iT^{2} \) |
| 11 | \( 1 + 6.26iT - 11T^{2} \) |
| 13 | \( 1 - 0.736T + 13T^{2} \) |
| 17 | \( 1 - 2.79iT - 17T^{2} \) |
| 19 | \( 1 + (-0.886 + 0.886i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + (-5.78 - 5.78i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.64 + 1.64i)T - 31iT^{2} \) |
| 41 | \( 1 - 0.847iT - 41T^{2} \) |
| 43 | \( 1 + 9.33T + 43T^{2} \) |
| 47 | \( 1 + (-2.86 - 2.86i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.98 + 3.98i)T - 53iT^{2} \) |
| 59 | \( 1 + (8.40 - 8.40i)T - 59iT^{2} \) |
| 61 | \( 1 + (5.05 - 5.05i)T - 61iT^{2} \) |
| 67 | \( 1 + (-6.24 + 6.24i)T - 67iT^{2} \) |
| 71 | \( 1 + 0.962T + 71T^{2} \) |
| 73 | \( 1 + (-2.59 - 2.59i)T + 73iT^{2} \) |
| 79 | \( 1 + (-11.6 + 11.6i)T - 79iT^{2} \) |
| 83 | \( 1 + (7.04 - 7.04i)T - 83iT^{2} \) |
| 89 | \( 1 + (12.7 + 12.7i)T + 89iT^{2} \) |
| 97 | \( 1 + 18.3iT - 97T^{2} \) |
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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
−10.20054755795021919329768503007, −8.855044903397868887837172215031, −8.164355275440488858711177802068, −7.04966886839190459940265838695, −6.31691712380229876887960039099, −5.74761582924125365190706994208, −4.87634296976606293296855393591, −3.39397375897168750239146720806, −2.76427161089835687927621276131, −1.30372634828802096506178221040,
1.20573974053725390927537128987, 2.47812679364085624150991985956, 4.02253282930735815268483782235, 4.80711834702942292862164526443, 5.13054263310204889536803799830, 6.35640038965633446511576399558, 7.18060508475163041440761991510, 8.080966078303916713425979231388, 9.277463574652826034890796261113, 9.829006865079595616226856811212