| L(s) = 1 | + i·2-s − 2i·3-s − 4-s + 2·6-s + i·7-s − i·8-s − 9-s + 3·11-s + 2i·12-s + i·13-s − 14-s + 16-s − 3i·17-s − i·18-s + 4·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 1.15i·3-s − 0.5·4-s + 0.816·6-s + 0.377i·7-s − 0.353i·8-s − 0.333·9-s + 0.904·11-s + 0.577i·12-s + 0.277i·13-s − 0.267·14-s + 0.250·16-s − 0.727i·17-s − 0.235i·18-s + 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46025 - 0.344719i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46025 - 0.344719i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 - 15T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 15iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−10.38647607678954729037297434716, −9.299524572331334031166434606079, −8.639590483965322316017607126919, −7.62286265732392386231499567223, −6.93543893936234043720817083710, −6.29476915092048769817889466014, −5.24324181032387123568365639283, −4.03456085125776766304225770892, −2.47084449387157313813355125967, −0.986160448718683558439772957054,
1.39959178083319643891731881799, 3.21734294158814495828468795182, 3.92209598822427649126366876776, 4.80048871964218044716275128104, 5.82910499375873064003413852779, 7.14855545326978542343110637644, 8.279021715723857189517346236527, 9.309024566816859540508558013598, 9.789023746212829269236502858853, 10.54541634006610212501995118854
