L(s) = 1 | + (1 − 1.73i)5-s + (2 + 3.46i)11-s + (−1.5 + 2.59i)13-s − 7·17-s + 5·19-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + (−0.5 − 0.866i)29-s + (1.5 − 2.59i)31-s + 11·37-s + (−4.5 + 7.79i)41-s + (−2.5 − 4.33i)43-s + (1.5 + 2.59i)47-s − 3·53-s + 7.99·55-s + ⋯ |
L(s) = 1 | + (0.447 − 0.774i)5-s + (0.603 + 1.04i)11-s + (−0.416 + 0.720i)13-s − 1.69·17-s + 1.14·19-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + (−0.0928 − 0.160i)29-s + (0.269 − 0.466i)31-s + 1.80·37-s + (−0.702 + 1.21i)41-s + (−0.381 − 0.660i)43-s + (0.218 + 0.378i)47-s − 0.412·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.954241321\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.954241321\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.5 - 2.59i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.370797942546796386827833187161, −7.48606639060235296571686745126, −6.78832048669187998558108714226, −6.25410170534140642804682541463, −5.18347126176157555556413580739, −4.59909170594762317703112129073, −4.10420893926056183715639464175, −2.73767309853704406212777867966, −1.95555841327081148811512009705, −1.02388004701166418935355461429,
0.58134295081294809973308100194, 1.85237026015284267678644527330, 2.89157418228949035411097426309, 3.35273085562019715385227528332, 4.45572687579287553979953728423, 5.29031963061729056388367555075, 6.07806453427207003368585551338, 6.61706161940970894091614936721, 7.31149809979171135891637925314, 8.096432636143850653166488770206