| L(s) = 1 | − i·2-s − 4-s + i·7-s + i·8-s + 4·11-s − 2i·13-s + 14-s + 16-s − 6i·17-s − 4i·22-s + 8i·23-s − 2·26-s − i·28-s + 10·29-s − 8·31-s − i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.377i·7-s + 0.353i·8-s + 1.20·11-s − 0.554i·13-s + 0.267·14-s + 0.250·16-s − 1.45i·17-s − 0.852i·22-s + 1.66i·23-s − 0.392·26-s − 0.188i·28-s + 1.85·29-s − 1.43·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.851080526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.851080526\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.762485760516814276506416481264, −7.88531183134421312445158632349, −7.10919006723036228969870917074, −6.22404062695061627230518680314, −5.34101421649244286684639540018, −4.65236891331094227497641237193, −3.60074446134783753497241225253, −2.96796623091637426906950349012, −1.84880430456966005618753217806, −0.78461377901731977779196383475,
0.927936750808387808129664658783, 2.12109098891343464276410383978, 3.58378121644805737314205261849, 4.18004133837923729013381197563, 4.92487744699090108147806507829, 6.10994024603543478578020940917, 6.50124166124116258540171234323, 7.16640032452598025229564295112, 8.128339392213194433757424584729, 8.749491786643779186228267231118
