| L(s) = 1 | − 2i·2-s + i·3-s − 2·4-s + 2·6-s + i·7-s − 9-s + 5·11-s − 2i·12-s − i·13-s + 2·14-s − 4·16-s + 7i·17-s + 2i·18-s + 6·19-s − 21-s − 10i·22-s + ⋯ |
| L(s) = 1 | − 1.41i·2-s + 0.577i·3-s − 4-s + 0.816·6-s + 0.377i·7-s − 0.333·9-s + 1.50·11-s − 0.577i·12-s − 0.277i·13-s + 0.534·14-s − 16-s + 1.69i·17-s + 0.471i·18-s + 1.37·19-s − 0.218·21-s − 2.13i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46799 - 0.907267i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46799 - 0.907267i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 + 2iT - 2T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 17 | \( 1 - 7iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 - 5iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 11T + 89T^{2} \) |
| 97 | \( 1 - 11iT - 97T^{2} \) |
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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
−9.996394008838985257822741622651, −9.243254900933984545463546409236, −8.740472855889520499932547426011, −7.42827363512632460370386271467, −6.25274741289702926900474348398, −5.32632479725822535632326487500, −3.86281532014586325850132885485, −3.73116461641501034639507186577, −2.32357140714888001353167288909, −1.16527492879914064662368930022,
1.06561585539677549750123855956, 2.79294761687251196328087093584, 4.25303219057324838174535968171, 5.16828518296412863150588415303, 6.18490496518153754585792650511, 6.84709493670711635863691803818, 7.39288382163713073087305494813, 8.187482288672853866644552945813, 9.217628409188435316700700330816, 9.633176476545315607029227176635
