L(s) = 1 | − i·2-s + i·3-s − 4-s + (2 − i)5-s + 6-s − 2i·7-s + i·8-s − 9-s + (−1 − 2i)10-s − 6·11-s − i·12-s − 2i·13-s − 2·14-s + (1 + 2i)15-s + 16-s − 6i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.894 − 0.447i)5-s + 0.408·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s + (−0.316 − 0.632i)10-s − 1.80·11-s − 0.288i·12-s − 0.554i·13-s − 0.534·14-s + (0.258 + 0.516i)15-s + 0.250·16-s − 1.45i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.222207 - 0.941287i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.222207 - 0.941287i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−9.920692181020961282739520091759, −9.114882570523260308065727861260, −8.252289531777121079229109701580, −7.32187816876209936881109139987, −5.92531900103016535393686587314, −5.09057196258572380238194577904, −4.45309952440051840068136497396, −3.07971093944410303561855635872, −2.19071424827379465570161767971, −0.41938997965137985629949346235,
1.97375645400248099072466309148, 2.77623432088683007099835644420, 4.47727261522209926947945209115, 5.53786055272347795924294184933, 6.21364179357907683663796927527, 6.78361127894603513312779486550, 8.082992993984181268186689906233, 8.423286872269524714293322793995, 9.458820960308478406152581506333, 10.51562922229609008700824318812