L(s) = 1 | + (0.224 + 0.224i)3-s + (−2 + 2i)7-s − 2.89i·9-s + 1.44i·11-s + (0.449 − 0.449i)13-s + (−3.22 − 3.22i)17-s + 7.44·19-s − 0.898·21-s + (4.44 + 4.44i)23-s + (1.32 − 1.32i)27-s + 6.89i·29-s − 6.89i·31-s + (−0.325 + 0.325i)33-s + (6 + 6i)37-s + 0.202·39-s + ⋯ |
L(s) = 1 | + (0.129 + 0.129i)3-s + (−0.755 + 0.755i)7-s − 0.966i·9-s + 0.437i·11-s + (0.124 − 0.124i)13-s + (−0.782 − 0.782i)17-s + 1.70·19-s − 0.196·21-s + (0.927 + 0.927i)23-s + (0.255 − 0.255i)27-s + 1.28i·29-s − 1.23i·31-s + (−0.0567 + 0.0567i)33-s + (0.986 + 0.986i)37-s + 0.0323·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.606311737\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.606311737\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.224 - 0.224i)T + 3iT^{2} \) |
| 7 | \( 1 + (2 - 2i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.44iT - 11T^{2} \) |
| 13 | \( 1 + (-0.449 + 0.449i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.22 + 3.22i)T + 17iT^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 + (-4.44 - 4.44i)T + 23iT^{2} \) |
| 29 | \( 1 - 6.89iT - 29T^{2} \) |
| 31 | \( 1 + 6.89iT - 31T^{2} \) |
| 37 | \( 1 + (-6 - 6i)T + 37iT^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + (-6.89 - 6.89i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.34 - 7.34i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.55 + 1.55i)T - 53iT^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 8.89T + 61T^{2} \) |
| 67 | \( 1 + (1.77 - 1.77i)T - 67iT^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 + (1.67 - 1.67i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.89T + 79T^{2} \) |
| 83 | \( 1 + (-2.67 - 2.67i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.10iT - 89T^{2} \) |
| 97 | \( 1 + (4.89 + 4.89i)T + 97iT^{2} \) |
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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
−9.325402135278724350398210272502, −9.135606373669327593607658661667, −7.83194617432971696827508884586, −7.05991210154952832599858244919, −6.25587759262286638103069358365, −5.45913186955469515957402545657, −4.48717350748919592025804429003, −3.27833638958792849593598535698, −2.74300513435413772548713468376, −1.05993373866904518575775006589,
0.76892180741799599700589028155, 2.26011762162480818249902262632, 3.29892563119849109391621122070, 4.22168954388882450686365735309, 5.20223251348097933706948156460, 6.15442653768003564207914336442, 7.01681014844949655469328787070, 7.64538341121996552868649602300, 8.537736354519051240084189709281, 9.292450936820120908360094398138