L(s) = 1 | + (0.707 + 0.707i)2-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + i·9-s + (−0.5 − 0.866i)11-s + 1.00i·14-s + 1.00·16-s + (−0.707 + 0.707i)18-s + (0.258 − 0.965i)22-s + (1.22 + 1.22i)23-s − 1.73·29-s + (1.22 − 1.22i)37-s + (0.707 − 0.707i)43-s + 1.73i·46-s + 1.00i·49-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + i·9-s + (−0.5 − 0.866i)11-s + 1.00i·14-s + 1.00·16-s + (−0.707 + 0.707i)18-s + (0.258 − 0.965i)22-s + (1.22 + 1.22i)23-s − 1.73·29-s + (1.22 − 1.22i)37-s + (0.707 − 0.707i)43-s + 1.73i·46-s + 1.00i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.834156718\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.834156718\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 3 | \( 1 - iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 29 | \( 1 + 1.73T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.73T + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.330865646263503639342446441500, −8.621165114042985497015674131224, −7.53930285373722468638220253706, −7.38441931184965124635186822560, −5.85349076877299203543430489658, −5.61919456654012897519458307904, −4.91276445684855803119322942393, −3.96244378377304108966962676764, −2.71914410694411972638808892775, −1.54280749903564503693342539208,
1.35167576889874469789600800282, 2.52636182131268238317603368080, 3.43843928962929672479478311916, 4.41480899988231610462222055098, 4.78033574879896514024707478299, 5.94580431544503446278653209202, 7.07465548975926881244387055142, 7.61015984010501799676955317655, 8.479926116815671329354204981316, 9.393492694240929312183313925571