L(s) = 1 | + (−0.5 − 0.866i)3-s + (1.70 − 2.95i)5-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s − 2.58·13-s − 3.41·15-s + (−1.12 − 1.94i)17-s + (1.41 − 2.44i)19-s + (−3.82 + 6.63i)23-s + (−3.32 − 5.76i)25-s + 0.999·27-s − 6.82·29-s + (0.585 + 1.01i)31-s + (−0.999 + 1.73i)33-s + (2 − 3.46i)37-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.763 − 1.32i)5-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s − 0.717·13-s − 0.881·15-s + (−0.271 − 0.471i)17-s + (0.324 − 0.561i)19-s + (−0.798 + 1.38i)23-s + (−0.665 − 1.15i)25-s + 0.192·27-s − 1.26·29-s + (0.105 + 0.182i)31-s + (−0.174 + 0.301i)33-s + (0.328 − 0.569i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7826825498\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7826825498\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.70 + 2.95i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 + (1.12 + 1.94i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.82 - 6.63i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 + (-0.585 - 1.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.585 - 1.01i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.12 + 10.6i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.82 + 4.89i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.31T + 71T^{2} \) |
| 73 | \( 1 + (-6.94 - 12.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.82 + 11.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.31T + 83T^{2} \) |
| 89 | \( 1 + (7.12 - 12.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.58T + 97T^{2} \) |
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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
−8.579086840569816273382183709269, −7.81584983383766933112253633457, −7.06391596930339478492291804476, −6.04955579194452940198268365749, −5.30715942788641762204539344097, −4.95220607381280058497057720362, −3.66798169541418190122368433306, −2.34750664456240727557624162190, −1.47381991701470758275874543860, −0.25676550043747858011932985074,
1.92492230425792169912528662613, 2.69187251322358883402885896332, 3.69780783483002676894171138462, 4.66046665014611978795504974966, 5.59923126544920025429837741171, 6.29576108767144463025188781868, 6.95819867207953408081140516977, 7.75789300821929654610196035580, 8.722800178107585595780319189034, 9.813279528455887288189137822859