L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.965 + 0.258i)3-s − 1.00i·4-s + (−0.500 + 0.866i)6-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s − 1.73i·11-s + (0.258 + 0.965i)12-s − 1.00·16-s + (0.707 − 0.707i)17-s + (0.258 − 0.965i)18-s + i·19-s + (−1.22 − 1.22i)22-s + (0.866 + 0.500i)24-s + (−0.707 + 0.707i)27-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.965 + 0.258i)3-s − 1.00i·4-s + (−0.500 + 0.866i)6-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s − 1.73i·11-s + (0.258 + 0.965i)12-s − 1.00·16-s + (0.707 − 0.707i)17-s + (0.258 − 0.965i)18-s + i·19-s + (−1.22 − 1.22i)22-s + (0.866 + 0.500i)24-s + (−0.707 + 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0706 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0706 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9545856571\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9545856571\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - 1.73iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 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 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 89 | \( 1 + 1.73T + T^{2} \) |
| 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
−10.91322260979690947461219863022, −10.06659141973583876358417397174, −9.299362209779989681807293377490, −8.011116415661803315747559785968, −6.58895647036694121778813064625, −5.84010270023975185948784375521, −5.17075549484904348552318256331, −3.98257983847515790569620352270, −3.04905299221340423687937919644, −1.11543928520314058932352548067,
2.13344339935663428547069227099, 3.90200853071741372853193173707, 4.84279187873061813366787269754, 5.55123046536008970087221092949, 6.69908104857988673737869719666, 7.18326940290546465131870257934, 8.098780472366114890942246314833, 9.389317442594265527258093123686, 10.34623537395131639457702074505, 11.32732548520430801547996108605