L(s) = 1 | − 3i·3-s − 7i·7-s − 9·9-s + 54·11-s − 55i·13-s − 18i·17-s − 25·19-s − 21·21-s + 18i·23-s + 27i·27-s + 54·29-s + 271·31-s − 162i·33-s − 314i·37-s − 165·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 1.48·11-s − 1.17i·13-s − 0.256i·17-s − 0.301·19-s − 0.218·21-s + 0.163i·23-s + 0.192i·27-s + 0.345·29-s + 1.57·31-s − 0.854i·33-s − 1.39i·37-s − 0.677·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.017376315\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.017376315\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 7iT - 343T^{2} \) |
| 11 | \( 1 - 54T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 18iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 25T + 6.85e3T^{2} \) |
| 23 | \( 1 - 18iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 54T + 2.43e4T^{2} \) |
| 31 | \( 1 - 271T + 2.97e4T^{2} \) |
| 37 | \( 1 + 314iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 360T + 6.89e4T^{2} \) |
| 43 | \( 1 - 163iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 522iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 36iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 126T + 2.05e5T^{2} \) |
| 61 | \( 1 - 47T + 2.26e5T^{2} \) |
| 67 | \( 1 + 343iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.08e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.05e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 568T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.42e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 439iT - 9.12e5T^{2} \) |
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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
−9.043489549090572496222499366155, −8.227156027655877752073021621797, −7.45582428244387797958789774062, −6.59644822228975531186182986916, −5.95482858596549819677102579659, −4.79937358438954831285970096414, −3.78049889278053770911545713004, −2.78148908002658107576742104953, −1.44944424887702306612152415128, −0.52899772828367240462423674364,
1.21639446469872368141758830268, 2.40772034300162293554848723998, 3.70631398055409540235275342180, 4.34482991094341921657101712414, 5.32049161632393736882652333550, 6.51104638832887127008378911074, 6.81318786416026731162709429189, 8.424326200094274154299585098569, 8.721794412025526577769542282224, 9.704153871904488121429806842047