L(s) = 1 | + (0.258 + 0.965i)3-s + (−1.36 − 0.366i)5-s + (0.707 − 1.22i)7-s + (−0.866 + 0.499i)9-s + (−0.965 + 0.258i)11-s − 1.41i·15-s + 17-s + (0.707 − 0.707i)19-s + (1.36 + 0.366i)21-s + (0.866 + 0.5i)25-s + (−0.707 − 0.707i)27-s + (1.36 − 0.366i)29-s + (−0.499 − 0.866i)33-s + (−1.41 + 1.41i)35-s + (1 − i)37-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)3-s + (−1.36 − 0.366i)5-s + (0.707 − 1.22i)7-s + (−0.866 + 0.499i)9-s + (−0.965 + 0.258i)11-s − 1.41i·15-s + 17-s + (0.707 − 0.707i)19-s + (1.36 + 0.366i)21-s + (0.866 + 0.5i)25-s + (−0.707 − 0.707i)27-s + (1.36 − 0.366i)29-s + (−0.499 − 0.866i)33-s + (−1.41 + 1.41i)35-s + (1 − i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9913330859\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9913330859\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
good | 5 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + 2iT - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.093788900336968908153360916950, −8.259575829397810147392737738280, −7.61653722279494060975679177894, −7.38209424236448950712441071038, −5.76168038733358743829221864715, −4.79092920084560334509919178346, −4.39929408269194825350283338913, −3.63053056009307360320197710727, −2.68831454556611454287507962448, −0.77460166788853807489614845908,
1.24472928348990391198171011593, 2.71241113812846995856487903373, 3.09334713476777655467979853976, 4.40403391580102678992032547056, 5.47275407105679898859132467891, 6.05457429406912584562754578183, 7.22552046494091747072290677355, 7.85061233590423849708451210549, 8.165032845985163807867596521387, 8.856814026539743196501812317443