| L(s) = 1 | + i·7-s − 6·11-s + 4i·13-s − 6i·17-s − 2·19-s + 6·29-s − 10·31-s + 2i·37-s + 6·41-s + 4i·43-s − 49-s − 12i·53-s + 14·61-s − 4i·67-s − 6·71-s + ⋯ |
| L(s) = 1 | + 0.377i·7-s − 1.80·11-s + 1.10i·13-s − 1.45i·17-s − 0.458·19-s + 1.11·29-s − 1.79·31-s + 0.328i·37-s + 0.937·41-s + 0.609i·43-s − 0.142·49-s − 1.64i·53-s + 1.79·61-s − 0.488i·67-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.262041219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.262041219\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−7.974993749143955514348494516331, −7.27864900614843589322956216799, −6.68472230711466095013779907733, −5.75694998868896289625131497731, −5.06809646210375517438779108257, −4.57962017951305505505833440253, −3.45085974324065457683500676928, −2.58530176153702754606729741851, −2.00099877705111668925064305878, −0.45586460558548579626281711890,
0.68922434238440671713520378046, 2.01713772566379161559439994364, 2.80724755414600459599811911102, 3.66542247193884457469287440118, 4.46001566948391859295971699818, 5.45648175859613797717418998714, 5.71640828628891669085236793668, 6.73539172803285177306077442875, 7.56293888691250174193412738247, 8.037672269532777277215710453566
