L(s) = 1 | − i·3-s + 2.73i·5-s + 7-s − 9-s − 4.73i·11-s + 1.46i·13-s + 2.73·15-s + 6.19·17-s + 7.46i·19-s − i·21-s + 6.73·23-s − 2.46·25-s + i·27-s + 3.46i·29-s − 2·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.22i·5-s + 0.377·7-s − 0.333·9-s − 1.42i·11-s + 0.406i·13-s + 0.705·15-s + 1.50·17-s + 1.71i·19-s − 0.218i·21-s + 1.40·23-s − 0.492·25-s + 0.192i·27-s + 0.643i·29-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58715 + 0.208953i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58715 + 0.208953i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 2.73iT - 5T^{2} \) |
| 11 | \( 1 + 4.73iT - 11T^{2} \) |
| 13 | \( 1 - 1.46iT - 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 - 7.46iT - 19T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 4.92iT - 37T^{2} \) |
| 41 | \( 1 - 7.66T + 41T^{2} \) |
| 43 | \( 1 + 0.535iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 1.07iT - 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 6iT - 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 8.53T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 + 5.46iT - 83T^{2} \) |
| 89 | \( 1 - 3.26T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−10.72787631171736087433988076918, −9.817435314123745653039673193675, −8.602092912672501638910861020232, −7.85641048762091714451602267156, −7.03589644736988787639717793448, −6.13161266203420683218625201564, −5.37868642932451268130022450942, −3.60952055061234962298056007745, −2.92734831132646356072402768093, −1.34675556196981808255592338859,
1.06202795561126622743477763683, 2.70585151279635278930060105283, 4.23682197105787509626874587641, 4.91338545921509558595985555953, 5.58411069973399102272998092884, 7.13855907419891903546172382423, 7.88859930328761115531906121406, 9.050088037860489629581791935361, 9.378637888916219214855787862129, 10.39887208501168690121270690151