L(s) = 1 | − 2i·7-s − 6·11-s + 4i·13-s − 6i·17-s − 4·19-s + 6·29-s + 4·31-s + 8i·37-s + 8i·43-s + 3·49-s + 6i·53-s + 6·59-s + 2·61-s + 4i·67-s + 12·71-s + ⋯ |
L(s) = 1 | − 0.755i·7-s − 1.80·11-s + 1.10i·13-s − 1.45i·17-s − 0.917·19-s + 1.11·29-s + 0.718·31-s + 1.31i·37-s + 1.21i·43-s + 0.428·49-s + 0.824i·53-s + 0.781·59-s + 0.256·61-s + 0.488i·67-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.100116542\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.100116542\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.455247075133481006299115217412, −8.049278148922435234064451192165, −7.07565868062094417145466418044, −6.70572607022142048531350673876, −5.60122303682821828430940295754, −4.73529025010744677752477421678, −4.30164505802317307492007249569, −2.97296350566869200410881877135, −2.37794142916099814416965707949, −0.934917132768772682642972601934,
0.38450013899625558583774129159, 2.06962935183071438755445512505, 2.69294332589138691533521372601, 3.66836286189505578103644705043, 4.74090596514049221835074435982, 5.52522631346523749459457208012, 5.96426032676046222619033011623, 6.94788494901748043621824269016, 8.024423368007723938671909323732, 8.220724802447683652948001314610