L(s) = 1 | − 3i·3-s − 3i·5-s − 2·7-s − 6·9-s + i·11-s − 9·15-s − 6·17-s + 4i·19-s + 6i·21-s + 23-s − 4·25-s + 9i·27-s + 8i·29-s + 7·31-s + 3·33-s + ⋯ |
L(s) = 1 | − 1.73i·3-s − 1.34i·5-s − 0.755·7-s − 2·9-s + 0.301i·11-s − 2.32·15-s − 1.45·17-s + 0.917i·19-s + 1.30i·21-s + 0.208·23-s − 0.800·25-s + 1.73i·27-s + 1.48i·29-s + 1.25·31-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - iT \) |
good | 3 | \( 1 + 3iT - 3T^{2} \) |
| 5 | \( 1 + 3iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - iT - 59T^{2} \) |
| 61 | \( 1 + 4iT - 61T^{2} \) |
| 67 | \( 1 + 5iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + 16T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−8.232076787659667131500341637994, −7.25769700003516417665818433006, −6.72617148832644076232922190570, −5.99696449012798082260190639332, −5.18519311850200898399935449564, −4.22599600876556877852310622870, −2.96394313715142754796332552638, −1.92892948951150798705339609868, −1.13382715382671769874122575354, 0,
2.68842288540089207453113789010, 2.90433129053123688060996573854, 4.01781540679499535849568740583, 4.50911498268406571824744437431, 5.60783085293647579536044915749, 6.43468283680466291160908087225, 6.91261142085879767085708962018, 8.153501340627254110874418683838, 8.951792919656583942452391720098