L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.258 − 0.965i)5-s + 0.999i·8-s + (0.5 − 0.866i)9-s + (0.258 − 0.965i)10-s + 1.41i·13-s + (−0.5 + 0.866i)16-s + (1.22 − 0.707i)17-s + (0.866 − 0.499i)18-s + (0.707 − 0.707i)20-s + (−0.866 + 0.499i)25-s + (−0.707 + 1.22i)26-s + (−0.866 + 0.499i)32-s + 1.41·34-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.258 − 0.965i)5-s + 0.999i·8-s + (0.5 − 0.866i)9-s + (0.258 − 0.965i)10-s + 1.41i·13-s + (−0.5 + 0.866i)16-s + (1.22 − 0.707i)17-s + (0.866 − 0.499i)18-s + (0.707 − 0.707i)20-s + (−0.866 + 0.499i)25-s + (−0.707 + 1.22i)26-s + (−0.866 + 0.499i)32-s + 1.41·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.633887227\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.633887227\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.258 + 0.965i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 1.41iT - T^{2} \) |
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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
−10.18658745469103961427389383533, −9.177643510856615800598431261797, −8.632996631091027626169286125296, −7.47719298503520071576981861898, −6.91444579081446850308441434103, −5.85924694811451610154457008753, −4.98136208364463972541178582166, −4.15730358470410986281333019978, −3.35668466705718864119965099261, −1.66565511279663989273621241601,
1.71814407002169286595163026182, 3.01497576584863335499380343426, 3.62206610801279245905909710501, 4.90531547620996476097653958693, 5.66210368692910213632213553871, 6.63691052031758747836875367480, 7.52785538438715164758391563587, 8.237062422799868176401919169810, 9.858288464859647191403501162665, 10.42626017482351401367824389534