L(s) = 1 | + i·2-s + 3-s − 4-s + 2i·5-s + i·6-s − 2i·7-s − i·8-s + 9-s − 2·10-s − 12-s + (−3 − 2i)13-s + 2·14-s + 2i·15-s + 16-s − 2·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.894i·5-s + 0.408i·6-s − 0.755i·7-s − 0.353i·8-s + 0.333·9-s − 0.632·10-s − 0.288·12-s + (−0.832 − 0.554i)13-s + 0.534·14-s + 0.516i·15-s + 0.250·16-s − 0.485·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.911703 + 0.487928i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.911703 + 0.487928i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + (3 + 2i)T \) |
good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−14.81502284481490964825941282866, −13.74217914458887684779145517637, −12.91154339683631396817715304873, −11.12830875965079958923609098612, −10.08531224531131389879503994586, −8.851745622938911644148999350723, −7.41831476835950984450226175229, −6.79786971902108167486082243354, −4.85249707345187774716120514420, −3.11282312486137849791066275163,
2.16401043466288981441025505835, 4.09590559657322186514860263713, 5.54761891959354108169770988651, 7.66415074492954259569590107456, 8.984026452504948289969072761145, 9.511202414981599377293774012324, 11.10594759747666845413619050874, 12.35987154513940471821027925503, 12.89700338987594582710171365238, 14.21173017908092884179812633889