| L(s) = 1 | + i·7-s + 3·9-s − 4·11-s + 2i·13-s + 6i·17-s − 8·19-s − 6·29-s + 8·31-s + 2i·37-s + 2·41-s − 4i·43-s + 8i·47-s − 49-s + 6i·53-s − 6·61-s + ⋯ |
| L(s) = 1 | + 0.377i·7-s + 9-s − 1.20·11-s + 0.554i·13-s + 1.45i·17-s − 1.83·19-s − 1.11·29-s + 1.43·31-s + 0.328i·37-s + 0.312·41-s − 0.609i·43-s + 1.16i·47-s − 0.142·49-s + 0.824i·53-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.026896981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.026896981\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.992903094702103705351172268204, −8.919330382079513576548056115159, −8.243724306143163975356336755463, −7.47011285572364215902889192047, −6.47758094124440516270583930495, −5.81206519022109189319313440131, −4.61039410653294145651867066362, −4.01828761299910274334162031536, −2.59723362413832936647261406193, −1.64670023248576994942418186967,
0.39778913001841796688175152024, 2.03056861323365657171738892390, 3.08334817946026054783807946778, 4.32377305846473301833053232353, 4.94747425022579641938684617095, 6.02224124161427072590741799890, 7.00781810551954968235323917709, 7.62987080601243796752423280141, 8.396385774894016392694733449916, 9.410916468241825229766145265603
