L(s) = 1 | − 1.61i·3-s − i·7-s − 1.61·9-s − i·11-s − 1.61·13-s − 17-s + 1.61i·19-s − 1.61·21-s + i·27-s − 29-s + 0.618i·31-s − 1.61·33-s + 2.61i·39-s + 41-s − i·43-s + ⋯ |
L(s) = 1 | − 1.61i·3-s − i·7-s − 1.61·9-s − i·11-s − 1.61·13-s − 17-s + 1.61i·19-s − 1.61·21-s + i·27-s − 29-s + 0.618i·31-s − 1.61·33-s + 2.61i·39-s + 41-s − i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5910114934\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5910114934\) |
\(L(1)\) |
|
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 | \( 1 + 1.61iT - T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 + 1.61T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - 1.61iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - 0.618iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 - 0.618iT - T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 + 0.618iT - T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 + 0.618iT - T^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 - 0.618T + T^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + 0.618T + T^{2} \) |
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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
−7.73858769749365510443121234510, −7.60686784246973024753074085497, −6.81196306070889522316269909129, −6.16307091778900805176193829291, −5.40817734288491161454754442675, −4.32464146740364387940538477509, −3.35866264594108745200205293175, −2.34021355019288633650008374236, −1.52140698720427526983637471325, −0.30685085800694714946030010550,
2.35624710858523781555697740030, 2.65755440860477085356212857283, 3.98671781572133616016195783297, 4.68061046561209791955590972037, 5.05116724938903643651891765014, 5.87338519582092453275473412395, 6.94418992391571277579815186616, 7.60519233706503418217431995352, 8.785402740278309147073546452890, 9.189046883754691625837819802678