L(s) = 1 | + (1.87 − 1.87i)3-s + (2.44 − 2.44i)7-s − 4i·9-s + 4.58·11-s + (−3.74 − 3.74i)13-s + (1.22 + 1.22i)17-s + 4.58i·19-s − 9.16i·21-s + (−4.89 − 4.89i)23-s + (−1.87 − 1.87i)27-s + 9.16·29-s + 4i·31-s + (8.57 − 8.57i)33-s + (−3.74 + 3.74i)37-s − 14·39-s + ⋯ |
L(s) = 1 | + (1.08 − 1.08i)3-s + (0.925 − 0.925i)7-s − 1.33i·9-s + 1.38·11-s + (−1.03 − 1.03i)13-s + (0.297 + 0.297i)17-s + 1.05i·19-s − 1.99i·21-s + (−1.02 − 1.02i)23-s + (−0.360 − 0.360i)27-s + 1.70·29-s + 0.718i·31-s + (1.49 − 1.49i)33-s + (−0.615 + 0.615i)37-s − 2.24·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.832652705\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.832652705\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.87 + 1.87i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.44 + 2.44i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.58T + 11T^{2} \) |
| 13 | \( 1 + (3.74 + 3.74i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.22 - 1.22i)T + 17iT^{2} \) |
| 19 | \( 1 - 4.58iT - 19T^{2} \) |
| 23 | \( 1 + (4.89 + 4.89i)T + 23iT^{2} \) |
| 29 | \( 1 - 9.16T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + (3.74 - 3.74i)T - 37iT^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + (-3.74 + 3.74i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.44 - 2.44i)T - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 9.16iT - 61T^{2} \) |
| 67 | \( 1 + (-1.87 - 1.87i)T + 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (-6.12 + 6.12i)T - 73iT^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + (5.61 - 5.61i)T - 83iT^{2} \) |
| 89 | \( 1 + 15iT - 89T^{2} \) |
| 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
−8.835337667692930045883990431261, −8.243498396939752468140236262940, −7.73921062392331173367383084049, −6.97775254503110747074231552295, −6.26661790633094286589662440657, −4.92030925393864784979346579709, −3.96610219142896494689486766972, −3.01743804301769756400686321204, −1.84207694329417953052114191498, −1.04696632069517372165952873529,
1.80119643492748554640486378129, 2.65267843713811435370226091681, 3.76351824271554312131644089288, 4.55398069894083476951621578803, 5.16432031978827419902718166204, 6.43736438235842738304121688362, 7.38780094688437059782774605313, 8.419728770593743757982171228630, 8.834780964613549454707553109152, 9.608782029955856412090357519063