| L(s) = 1 | − i·3-s − 2i·7-s + 2·9-s − 4·11-s − 5i·13-s + 2i·17-s − 6·19-s − 2·21-s + i·23-s − 5i·27-s − 29-s − 9·31-s + 4i·33-s + 4i·37-s − 5·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.755i·7-s + 0.666·9-s − 1.20·11-s − 1.38i·13-s + 0.485i·17-s − 1.37·19-s − 0.436·21-s + 0.208i·23-s − 0.962i·27-s − 0.185·29-s − 1.61·31-s + 0.696i·33-s + 0.657i·37-s − 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 9T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 5iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 + 15iT - 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−7.72192308001022868304892860098, −7.35056794366390380867642246525, −6.38055219821832046866988926071, −5.74983199475273923966557123214, −4.82381956575589086694887641647, −4.07650839955192711372972108393, −3.12792862105891749147473144489, −2.18597920951845860254381719540, −1.15946400338431335773317452442, 0,
1.88076293422317677242153232157, 2.42637272773495874480543280662, 3.68150592729800116861449404187, 4.32320847214229393609346927889, 5.11782060213045759144569176827, 5.71531811598472010858017923047, 6.74461630248785559806007287921, 7.27122553736599112400227658848, 8.175229023606042385992057234139
