L(s) = 1 | + (−0.673 + 0.118i)5-s + (0.766 + 0.642i)9-s + (1.11 + 1.32i)13-s + (1.26 − 1.50i)17-s + (−0.500 + 0.181i)25-s + (−1.11 − 0.642i)29-s + (−0.766 + 0.642i)37-s + (−0.266 + 0.223i)41-s + (−0.592 − 0.342i)45-s + (−0.939 + 0.342i)49-s + (0.173 − 0.984i)53-s + (−0.826 − 0.984i)61-s + (−0.907 − 0.761i)65-s + 73-s + (0.173 + 0.984i)81-s + ⋯ |
L(s) = 1 | + (−0.673 + 0.118i)5-s + (0.766 + 0.642i)9-s + (1.11 + 1.32i)13-s + (1.26 − 1.50i)17-s + (−0.500 + 0.181i)25-s + (−1.11 − 0.642i)29-s + (−0.766 + 0.642i)37-s + (−0.266 + 0.223i)41-s + (−0.592 − 0.342i)45-s + (−0.939 + 0.342i)49-s + (0.173 − 0.984i)53-s + (−0.826 − 0.984i)61-s + (−0.907 − 0.761i)65-s + 73-s + (0.173 + 0.984i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8950703145\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8950703145\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 + (0.766 - 0.642i)T \) |
good | 3 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 5 | \( 1 + (0.673 - 0.118i)T + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.11 - 1.32i)T + (-0.173 + 0.984i)T^{2} \) |
| 17 | \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \) |
| 19 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (0.826 + 0.984i)T + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 89 | \( 1 + (1.93 + 0.342i)T + (0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \) |
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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
−11.19999509415452138682016670002, −9.979762501229987188374107141856, −9.317291420422317572628053084852, −8.149954891650736220485821897350, −7.44107899532965075407921531019, −6.61419294587995067397108551881, −5.33063779785418827055093916835, −4.29854597402642280072217352899, −3.36209230845562546230434852951, −1.69428104488124647492310375614,
1.36316111834038592372152018835, 3.45118103062819839022177575889, 3.91980819586604418465150185011, 5.44367354334518441885365385300, 6.24885655882283037685613687615, 7.49252975332616077752233453450, 8.110829825287575311052649244827, 9.016933298363339119305496902966, 10.17867218968191143833308191308, 10.69011055671439880650225803600