| L(s) = 1 | − 2-s + i·3-s + 4-s − i·6-s + 3.12·7-s − 8-s − 9-s + 5.12i·11-s + i·12-s + (0.561 + 3.56i)13-s − 3.12·14-s + 16-s − 2i·17-s + 18-s + 6i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.5·4-s − 0.408i·6-s + 1.18·7-s − 0.353·8-s − 0.333·9-s + 1.54i·11-s + 0.288i·12-s + (0.155 + 0.987i)13-s − 0.834·14-s + 0.250·16-s − 0.485i·17-s + 0.235·18-s + 1.37i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.180151944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.180151944\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-0.561 - 3.56i)T \) |
| good | 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 - 5.12iT - 11T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 5.12iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 3.12iT - 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 - 0.876iT - 41T^{2} \) |
| 43 | \( 1 - 6.24iT - 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 13.3iT - 53T^{2} \) |
| 59 | \( 1 - 1.12iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 4.87T + 67T^{2} \) |
| 71 | \( 1 - 10.2iT - 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 3.12iT - 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−9.685887977641058566095011105584, −8.574426917416526637114467052008, −8.145057629330969187166984129762, −7.18816333573932732179343715331, −6.53178135869764567401411835335, −5.26028132892210687954445313447, −4.63463139422776371304037847504, −3.76299799816882694576718587296, −2.26272141564055991122881952317, −1.54247966423304117161995704142,
0.55184023070472585412947222210, 1.56846233603366561074380627698, 2.72749167829399010920066429819, 3.69960303142110675603502089834, 5.18417027208865692373149661992, 5.72622434175477454554832794838, 6.69910724033158905015173591054, 7.60649812270341498530079491582, 8.176664550372164133919940557089, 8.678677933958750871834344566111
