L(s) = 1 | + (2.59 − 1.5i)5-s + (1.5 + 0.866i)7-s + (−2.59 + 4.5i)11-s + (1 + 1.73i)13-s + 6i·17-s + 6.92i·19-s + (2 − 3.46i)25-s + (−5.19 − 3i)29-s + (−4.5 + 2.59i)31-s + 5.19·35-s + 8·37-s + (9 + 5.19i)43-s + (5.19 − 9i)47-s + (−2 − 3.46i)49-s − 9i·53-s + ⋯ |
L(s) = 1 | + (1.16 − 0.670i)5-s + (0.566 + 0.327i)7-s + (−0.783 + 1.35i)11-s + (0.277 + 0.480i)13-s + 1.45i·17-s + 1.58i·19-s + (0.400 − 0.692i)25-s + (−0.964 − 0.557i)29-s + (−0.808 + 0.466i)31-s + 0.878·35-s + 1.31·37-s + (1.37 + 0.792i)43-s + (0.757 − 1.31i)47-s + (−0.285 − 0.494i)49-s − 1.23i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.994080772\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.994080772\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.59 + 1.5i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.59 - 4.5i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 6.92iT - 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.19 + 3i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-9 - 5.19i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.19 + 9i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + (5.19 + 9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + (-3 - 1.73i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.59 + 4.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.818941360434457394457440965614, −9.028596266009866566594451705895, −8.177753286158167706664971688816, −7.47845702382015483374178690110, −6.17863368502828236627710621693, −5.63870512433124292640436660338, −4.78754293319957480242698568406, −3.83465941898381999698032927350, −2.02752263734196660047599907058, −1.73932551197455265844130717145,
0.837075201525958871007728023307, 2.48232884004690307500733392751, 3.02325865326981425920612102553, 4.52438560038646191763743599519, 5.57209092657680689699535401521, 5.99233258507094336906712548710, 7.21130812373886920530098881443, 7.72253084523371461803828688607, 9.018585815358247672465948666252, 9.363587045346512442658567819164