L(s) = 1 | − 1.41i·2-s − 1.00·4-s + (0.707 − 0.707i)5-s + i·7-s + (−1.00 − 1.00i)10-s + (0.707 − 0.707i)11-s − i·13-s + 1.41·14-s − 0.999·16-s + (0.707 + 0.707i)17-s + (1 + i)19-s + (−0.707 + 0.707i)20-s + (−1.00 − 1.00i)22-s − 1.00i·25-s − 1.41·26-s + ⋯ |
L(s) = 1 | − 1.41i·2-s − 1.00·4-s + (0.707 − 0.707i)5-s + i·7-s + (−1.00 − 1.00i)10-s + (0.707 − 0.707i)11-s − i·13-s + 1.41·14-s − 0.999·16-s + (0.707 + 0.707i)17-s + (1 + i)19-s + (−0.707 + 0.707i)20-s + (−1.00 − 1.00i)22-s − 1.00i·25-s − 1.41·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.336626824\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.336626824\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 + 1.41iT - T^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 17 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 19 | \( 1 + (-1 - i)T + iT^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (1 - i)T - iT^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−9.350929186516249797348976650960, −8.736347765252392497781097641888, −8.042483528471385802420204745633, −6.62833677174338821874652659025, −5.60543178401298947208841472014, −5.24833570867800876940961912810, −3.75176429716497299987928368642, −3.17989832596480870805830599624, −1.98542069498423689015655094144, −1.18542602498419571904832543431,
1.67258299556522350947445724922, 3.05721606575257696380895749907, 4.30629530587323580540908815889, 5.05156937654139062404948555441, 6.04284074678674949896700913563, 6.76751116023745763093047111594, 7.24316361784957110669837768702, 7.73093602563043646570462909576, 9.130541879686204711163506428462, 9.459401298911257458510849032200