| L(s) = 1 | − i·2-s − 4-s − 4i·7-s + i·8-s + 3·11-s + i·13-s − 4·14-s + 16-s − 2·19-s − 3i·22-s − 9i·23-s + 26-s + 4i·28-s + 6·29-s − 10·31-s − i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.51i·7-s + 0.353i·8-s + 0.904·11-s + 0.277i·13-s − 1.06·14-s + 0.250·16-s − 0.458·19-s − 0.639i·22-s − 1.87i·23-s + 0.196·26-s + 0.755i·28-s + 1.11·29-s − 1.79·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.305066015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.305066015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 19iT - 97T^{2} \) |
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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
−9.310197694312992821335818013624, −8.676968055045021188693936347851, −7.61739561558074956814453841083, −6.86727380883049595868492689978, −6.01408839893684542016735533796, −4.48196000563573412167311669178, −4.21723998255632800285921148902, −3.11050540569424984993948943883, −1.75211956586674351532443810815, −0.55639531430636404842702772651,
1.61508548347768100118048311247, 2.96606982920080242239394955453, 4.04176320720505448949022619645, 5.22593223358692781622061061921, 5.79178397492372054031604074818, 6.57988672860641656954622363287, 7.52045503031765583753327774724, 8.388506319583314764017834104905, 9.133852585253416312549552097848, 9.509336493224992607921097700515
