L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−1.62 + 2.09i)7-s + (−0.499 − 0.866i)9-s + (−2.12 + 3.67i)11-s + 3.24·13-s + 0.999·15-s + (−2.12 + 3.67i)17-s + (−3.5 − 6.06i)19-s + (0.999 + 2.44i)21-s + (−2.12 − 3.67i)23-s + (−0.499 + 0.866i)25-s − 0.999·27-s − 1.75·29-s + (−4.74 + 8.21i)31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.223 + 0.387i)5-s + (−0.612 + 0.790i)7-s + (−0.166 − 0.288i)9-s + (−0.639 + 1.10i)11-s + 0.899·13-s + 0.258·15-s + (−0.514 + 0.891i)17-s + (−0.802 − 1.39i)19-s + (0.218 + 0.534i)21-s + (−0.442 − 0.766i)23-s + (−0.0999 + 0.173i)25-s − 0.192·27-s − 0.326·29-s + (−0.851 + 1.47i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7094725656\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7094725656\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (1.62 - 2.09i)T \) |
good | 11 | \( 1 + (2.12 - 3.67i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.24T + 13T^{2} \) |
| 17 | \( 1 + (2.12 - 3.67i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.12 + 3.67i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.75T + 29T^{2} \) |
| 31 | \( 1 + (4.74 - 8.21i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.62 + 2.80i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 3.24T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.24 - 7.34i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.12 - 8.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.24 + 3.88i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.62 - 4.54i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + (4.62 - 8.00i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + (-5.12 - 8.87i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.485T + 97T^{2} \) |
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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
−9.517526980740991201464557942073, −8.788160573508715414694400675013, −8.242780864949757746398277120197, −7.02116535960512809784683505371, −6.63184565245481584152736115493, −5.76975472520627576144018221744, −4.74854889558621941820271191802, −3.58425650033496315336690991275, −2.54942453819010427698374911348, −1.84197818406225752534186192206,
0.23761505732460834420497445216, 1.83808710557813370129038928978, 3.29608244239695039311081512754, 3.78438793808603064515162068640, 4.86681044233011888321634773242, 5.86225775703433962826963990304, 6.43888056740140074101200871595, 7.74703148828337891491615907407, 8.202474875641004376016605506995, 9.182708807168847205953428027059