L(s) = 1 | + (−0.254 + 1.71i)3-s − 2i·5-s − 0.508i·7-s + (−2.87 − 0.870i)9-s + 4.44·11-s − 1.74·13-s + (3.42 + 0.508i)15-s − i·17-s − 1.01i·19-s + (0.870 + 0.129i)21-s + 6.34·23-s + 25-s + (2.22 − 4.69i)27-s − 9.48i·29-s − 4.44i·31-s + ⋯ |
L(s) = 1 | + (−0.146 + 0.989i)3-s − 0.894i·5-s − 0.192i·7-s + (−0.956 − 0.290i)9-s + 1.33·11-s − 0.483·13-s + (0.884 + 0.131i)15-s − 0.242i·17-s − 0.233i·19-s + (0.190 + 0.0281i)21-s + 1.32·23-s + 0.200·25-s + (0.427 − 0.903i)27-s − 1.76i·29-s − 0.798i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48048 - 0.109204i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48048 - 0.109204i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.254 - 1.71i)T \) |
| 17 | \( 1 + iT \) |
good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + 0.508iT - 7T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 19 | \( 1 + 1.01iT - 19T^{2} \) |
| 23 | \( 1 - 6.34T + 23T^{2} \) |
| 29 | \( 1 + 9.48iT - 29T^{2} \) |
| 31 | \( 1 + 4.44iT - 31T^{2} \) |
| 37 | \( 1 - 5.48T + 37T^{2} \) |
| 41 | \( 1 - 5.48iT - 41T^{2} \) |
| 43 | \( 1 - 8.88iT - 43T^{2} \) |
| 47 | \( 1 - 5.83T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 + 9.48T + 61T^{2} \) |
| 67 | \( 1 - 2.03iT - 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 9.26iT - 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 9.22iT - 89T^{2} \) |
| 97 | \( 1 - 5.48T + 97T^{2} \) |
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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
−9.978609775298164939477984329537, −9.376551001462741113238639659220, −8.845030862333051230269676974020, −7.81846126232491554652535802120, −6.60116324389441015021488662541, −5.67015151590990511968826019041, −4.60499102757659859498268250130, −4.15197535993323590249020145094, −2.78792080227906531202727363667, −0.901309006771423718682256954882,
1.28539054486414312188059157770, 2.59090947288932995415372462757, 3.58555468279948300257321051784, 5.09073062836971926166870291957, 6.12429213559893383685661684516, 6.98926218341337581369807991425, 7.27117971523283278880334112131, 8.655793211925467752318702596252, 9.172989370636345836379824649187, 10.56227300114950616600979239219