| L(s) = 1 | + (−2.13 − 0.656i)5-s + 3.46i·7-s − 11-s − 6.09i·13-s + 3.46i·17-s − 4·19-s − 8.24i·23-s + (4.13 + 2.80i)25-s + 6.54·29-s + 1.72·31-s + (2.27 − 7.40i)35-s + 8.24i·37-s + 6.54·41-s − 3.46i·43-s + 2.62i·47-s + ⋯ |
| L(s) = 1 | + (−0.955 − 0.293i)5-s + 1.30i·7-s − 0.301·11-s − 1.68i·13-s + 0.840i·17-s − 0.917·19-s − 1.71i·23-s + (0.827 + 0.561i)25-s + 1.21·29-s + 0.309·31-s + (0.384 − 1.25i)35-s + 1.35i·37-s + 1.02·41-s − 0.528i·43-s + 0.383i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.238342920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.238342920\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.13 + 0.656i)T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 13 | \( 1 + 6.09iT - 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8.24iT - 23T^{2} \) |
| 29 | \( 1 - 6.54T + 29T^{2} \) |
| 31 | \( 1 - 1.72T + 31T^{2} \) |
| 37 | \( 1 - 8.24iT - 37T^{2} \) |
| 41 | \( 1 - 6.54T + 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 2.62iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 + 10.8iT - 67T^{2} \) |
| 71 | \( 1 - 2.27T + 71T^{2} \) |
| 73 | \( 1 - 11.3iT - 73T^{2} \) |
| 79 | \( 1 + 4.54T + 79T^{2} \) |
| 83 | \( 1 + 1.78iT - 83T^{2} \) |
| 89 | \( 1 - 8.27T + 89T^{2} \) |
| 97 | \( 1 + 3.94iT - 97T^{2} \) |
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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
−8.665950263193901166281959554682, −8.469662346869528255026228903010, −7.898989069643527328993193231322, −6.68549744004515544790151826005, −5.92200717968363549969957210353, −5.07937436591063027038819805426, −4.28971650097760215770118765459, −3.11784926045600808394062137112, −2.39234939113654450245573041633, −0.64857501351959229389440771382,
0.833198583990112275557142562050, 2.31204908252880802627129405536, 3.61632316873437119713778267891, 4.17064388666367297166876675638, 4.90179571357458447800173824358, 6.26452187948975360791547254144, 7.16603091020996261528940503824, 7.35089954295982990662995623883, 8.355792834651609517074743475361, 9.194804708740832567175564266315
