| L(s) = 1 | + (−1 + i)3-s + (1 + i)7-s + i·9-s + 4i·11-s + (4 + 4i)13-s + (4 − 4i)17-s − 4·19-s − 2·21-s + (5 − 5i)23-s + (−4 − 4i)27-s − 2i·29-s + 8i·31-s + (−4 − 4i)33-s − 8·39-s − 4·41-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.577i)3-s + (0.377 + 0.377i)7-s + 0.333i·9-s + 1.20i·11-s + (1.10 + 1.10i)13-s + (0.970 − 0.970i)17-s − 0.917·19-s − 0.436·21-s + (1.04 − 1.04i)23-s + (−0.769 − 0.769i)27-s − 0.371i·29-s + 1.43i·31-s + (−0.696 − 0.696i)33-s − 1.28·39-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.354679712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.354679712\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1 - i)T + 7iT^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + (-4 - 4i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4 + 4i)T - 17iT^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-5 + 5i)T - 23iT^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + (7 - 7i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3 - 3i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4 - 4i)T + 53iT^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (-3 - 3i)T + 67iT^{2} \) |
| 71 | \( 1 - 16iT - 71T^{2} \) |
| 73 | \( 1 + (4 + 4i)T + 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (-5 + 5i)T - 83iT^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 + (12 - 12i)T - 97iT^{2} \) |
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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
−9.799213305508140722053793563820, −8.892890613355382179745385989393, −8.244102082229293383318882509651, −7.11515040985469529459204446923, −6.49187949885099807548182026998, −5.35600697270186469901385472790, −4.76900576626099330319206888718, −4.04886027243127850007050633577, −2.63970096018183434443584677962, −1.48924525087183715637078100899,
0.62363932292561094271429362883, 1.52306688034955979937522293970, 3.28228076936687235527126775812, 3.82921008566749396781685750606, 5.37907733006028821695365489587, 5.85807656234190864984834218509, 6.57925370113038588392895013871, 7.58955111222279669233963023116, 8.288093545460029554950883936100, 8.933674618950114818973548524953
