L(s) = 1 | − 2-s + 1.41i·3-s + 4-s + 3.74·5-s − 1.41i·6-s + 2.64i·7-s − 8-s + 0.999·9-s − 3.74·10-s + 1.41i·12-s − 5.65i·13-s − 2.64i·14-s + 5.29i·15-s + 16-s − 3.74·17-s − 0.999·18-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.816i·3-s + 0.5·4-s + 1.67·5-s − 0.577i·6-s + 0.999i·7-s − 0.353·8-s + 0.333·9-s − 1.18·10-s + 0.408i·12-s − 1.56i·13-s − 0.707i·14-s + 1.36i·15-s + 0.250·16-s − 0.907·17-s − 0.235·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12173 + 0.602952i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12173 + 0.602952i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 - 2.64iT \) |
| 23 | \( 1 + (-4 - 2.64i)T \) |
good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 5 | \( 1 - 3.74T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 3.74T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 1.41iT - 31T^{2} \) |
| 37 | \( 1 - 10.5iT - 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 5.29iT - 43T^{2} \) |
| 47 | \( 1 + 7.07iT - 47T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 9.89iT - 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 11.3iT - 73T^{2} \) |
| 79 | \( 1 + 10.5iT - 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 3.74T + 97T^{2} \) |
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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
−11.40817422789070904277784841156, −10.49673454548727677105610310345, −9.798113327217171748477516207869, −9.264112883301487643091746960160, −8.356604533998303811219512243228, −6.87323420803761577597016644613, −5.73449253747493668126872226590, −5.11190248178823588154160193147, −3.10003972301179746458859010178, −1.85258843189487934132622444710,
1.38766537805611112302199673759, 2.26255319679862247724222898727, 4.38188763453027629351225481002, 5.98996458540410725304662528252, 6.83759597024854198131915824309, 7.35889979155232871861819320709, 8.947371162707828213946523492231, 9.475800538349858567144590266948, 10.47961496570751210659668544719, 11.20716038088712338476969425459