L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.811 + 2.08i)5-s + (−3.03 + 3.03i)7-s − 1.00i·9-s + 0.215i·11-s + (−0.104 + 0.104i)13-s + (−2.04 − 0.899i)15-s + (−0.838 + 0.838i)17-s − 3.62·19-s − 4.28i·21-s + (1.87 + 4.41i)23-s + (−3.68 + 3.38i)25-s + (0.707 + 0.707i)27-s − 5.72i·29-s − 0.698·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.362 + 0.931i)5-s + (−1.14 + 1.14i)7-s − 0.333i·9-s + 0.0648i·11-s + (−0.0288 + 0.0288i)13-s + (−0.528 − 0.232i)15-s + (−0.203 + 0.203i)17-s − 0.832·19-s − 0.935i·21-s + (0.390 + 0.920i)23-s + (−0.736 + 0.676i)25-s + (0.136 + 0.136i)27-s − 1.06i·29-s − 0.125·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4979102428\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4979102428\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.811 - 2.08i)T \) |
| 23 | \( 1 + (-1.87 - 4.41i)T \) |
good | 7 | \( 1 + (3.03 - 3.03i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.215iT - 11T^{2} \) |
| 13 | \( 1 + (0.104 - 0.104i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.838 - 0.838i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.62T + 19T^{2} \) |
| 29 | \( 1 + 5.72iT - 29T^{2} \) |
| 31 | \( 1 + 0.698T + 31T^{2} \) |
| 37 | \( 1 + (-3.51 + 3.51i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.54T + 41T^{2} \) |
| 43 | \( 1 + (-1.20 - 1.20i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.22 + 3.22i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.708 + 0.708i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.79iT - 59T^{2} \) |
| 61 | \( 1 + 5.64iT - 61T^{2} \) |
| 67 | \( 1 + (5.33 - 5.33i)T - 67iT^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + (-9.19 + 9.19i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.634T + 79T^{2} \) |
| 83 | \( 1 + (-3.22 - 3.22i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.91T + 89T^{2} \) |
| 97 | \( 1 + (11.7 - 11.7i)T - 97iT^{2} \) |
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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
−9.894723348599403549218252530328, −9.488956905112277414786187201673, −8.661278382793318805466914775257, −7.48390323205879758286189213606, −6.46003554853379909509367540969, −6.09879053609311128299035539561, −5.25651183506350029416688854190, −3.93893073875073277224390667960, −3.02881600101299947915920288013, −2.13646539445015886141351099297,
0.21500584115852156018560241988, 1.34773615405423987050530699223, 2.82476635455439244207645583306, 4.08456323203344640347440192445, 4.82838654750630991279166006264, 5.93468281406274132808179960580, 6.62560870159789056352412865945, 7.31168417043640360261448445248, 8.369924820555330899222609656809, 9.109741009970871722330352976899