L(s) = 1 | − 2-s + i·3-s + 4-s + 3.74·5-s − i·6-s + (−1.87 − 1.87i)7-s − 8-s − 9-s − 3.74·10-s + i·12-s − 2i·13-s + (1.87 + 1.87i)14-s + 3.74i·15-s + 16-s + 3.74·17-s + 18-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.5·4-s + 1.67·5-s − 0.408i·6-s + (−0.707 − 0.707i)7-s − 0.353·8-s − 0.333·9-s − 1.18·10-s + 0.288i·12-s − 0.554i·13-s + (0.499 + 0.499i)14-s + 0.966i·15-s + 0.250·16-s + 0.907·17-s + 0.235·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46350 - 0.0802585i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46350 - 0.0802585i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (1.87 + 1.87i)T \) |
| 23 | \( 1 + (3 + 3.74i)T \) |
good | 5 | \( 1 - 3.74T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 3.74iT - 53T^{2} \) |
| 59 | \( 1 + 14iT - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 3.74iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 3.74iT - 79T^{2} \) |
| 83 | \( 1 + 7.48T + 83T^{2} \) |
| 89 | \( 1 - 3.74T + 89T^{2} \) |
| 97 | \( 1 - 7.48T + 97T^{2} \) |
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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
−9.970347278011428873344245702998, −9.553783873267372148624898558884, −8.542096522581709937828670534004, −7.55817556886872217289723200015, −6.44753041308674786111570931691, −5.89886067602748568576145236734, −4.90322194971579049509452102493, −3.42439309720576910154652819510, −2.46963275274178490827704973027, −0.990833876426807098094216900052,
1.30756390169699774593462849571, 2.27564002357585539047867005583, 3.21592604369683581621229018348, 5.25872452946625036902448071790, 5.92071392642968751950297531986, 6.57647885599191137458524477858, 7.45720161566668508631493321396, 8.613238624240921713650314802636, 9.275416069701885848509775220201, 9.882370751444589459994497672004