| L(s) = 1 | + 0.711·3-s + 0.887·5-s + 2.32i·7-s − 2.49·9-s − i·11-s + 0.0550i·13-s + 0.631·15-s − 0.537·17-s + (−3.30 − 2.83i)19-s + 1.65i·21-s − 2.34i·23-s − 4.21·25-s − 3.91·27-s − 5.27i·29-s − 4.62·31-s + ⋯ |
| L(s) = 1 | + 0.411·3-s + 0.396·5-s + 0.880i·7-s − 0.831·9-s − 0.301i·11-s + 0.0152i·13-s + 0.163·15-s − 0.130·17-s + (−0.759 − 0.650i)19-s + 0.361i·21-s − 0.488i·23-s − 0.842·25-s − 0.752·27-s − 0.980i·29-s − 0.829·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6710630867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6710630867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + iT \) |
| 19 | \( 1 + (3.30 + 2.83i)T \) |
| good | 3 | \( 1 - 0.711T + 3T^{2} \) |
| 5 | \( 1 - 0.887T + 5T^{2} \) |
| 7 | \( 1 - 2.32iT - 7T^{2} \) |
| 13 | \( 1 - 0.0550iT - 13T^{2} \) |
| 17 | \( 1 + 0.537T + 17T^{2} \) |
| 23 | \( 1 + 2.34iT - 23T^{2} \) |
| 29 | \( 1 + 5.27iT - 29T^{2} \) |
| 31 | \( 1 + 4.62T + 31T^{2} \) |
| 37 | \( 1 - 2.51iT - 37T^{2} \) |
| 41 | \( 1 + 12.4iT - 41T^{2} \) |
| 43 | \( 1 - 9.53iT - 43T^{2} \) |
| 47 | \( 1 + 4.22iT - 47T^{2} \) |
| 53 | \( 1 + 4.95iT - 53T^{2} \) |
| 59 | \( 1 + 6.86T + 59T^{2} \) |
| 61 | \( 1 + 7.13T + 61T^{2} \) |
| 67 | \( 1 - 3.31T + 67T^{2} \) |
| 71 | \( 1 - 8.62T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 + 3.87iT - 83T^{2} \) |
| 89 | \( 1 - 3.77iT - 89T^{2} \) |
| 97 | \( 1 + 17.5iT - 97T^{2} \) |
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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
−8.450224479114537156538225202046, −7.82699712310127056349277431521, −6.76193216547817601839008505573, −5.96493603708851436584976574971, −5.51928492490200737354750288088, −4.49419414127875448457284359942, −3.47511684505830973420159606677, −2.54225171542744745381768488364, −1.98457990092675374452763724064, −0.17462828907076925189600640830,
1.43870975256287622097559312698, 2.38564857402642256585778989321, 3.43495505657218092342900101344, 4.09709404366005325144640496823, 5.12553416719781235824095063793, 5.91904076002084920701294128796, 6.64429593692491760236342788526, 7.58841300417689261901229140141, 8.022171516243356797893552230131, 9.015371480125640304076919698781
