L(s) = 1 | + i·3-s + 5-s + i·11-s + i·15-s − 17-s + i·19-s − i·23-s + i·27-s + i·31-s − 33-s + 37-s − i·47-s − i·51-s + 53-s + i·55-s + ⋯ |
L(s) = 1 | + i·3-s + 5-s + i·11-s + i·15-s − 17-s + i·19-s − i·23-s + i·27-s + i·31-s − 33-s + 37-s − i·47-s − i·51-s + 53-s + i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.485652537\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.485652537\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - iT - T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 + iT - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - iT - T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + iT - T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - iT - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.241876990675967998884559216376, −8.562972428540158871598491415602, −7.52172645344500609831401430436, −6.69096785298671250857312614838, −5.99066298073151943228890797776, −5.05885138415428271470975104212, −4.49012310413206025673875988777, −3.70625648228680756649576216384, −2.48117009337847055691089189396, −1.65953217051652081927372094855,
0.940074708496143179824995466120, 2.02631818263215277663268418528, 2.70051482340089519087629051896, 3.94683850696249150388724611620, 4.98204821140025698768923604444, 5.96847750867158648624665783734, 6.30135256160612369572814490705, 7.16171680186417372324611151230, 7.83043602549637263353919826910, 8.672873226325139418633983288125