L(s) = 1 | + (−0.925 + 0.925i)3-s + (−0.707 − 0.707i)5-s − i·7-s + 1.28i·9-s + (−3.01 − 3.01i)11-s + (−0.350 + 0.350i)13-s + 1.30·15-s + 5.90·17-s + (−0.223 + 0.223i)19-s + (0.925 + 0.925i)21-s + 2.20i·23-s + 1.00i·25-s + (−3.96 − 3.96i)27-s + (1.25 − 1.25i)29-s − 2.03·31-s + ⋯ |
L(s) = 1 | + (−0.534 + 0.534i)3-s + (−0.316 − 0.316i)5-s − 0.377i·7-s + 0.428i·9-s + (−0.908 − 0.908i)11-s + (−0.0971 + 0.0971i)13-s + 0.338·15-s + 1.43·17-s + (−0.0512 + 0.0512i)19-s + (0.202 + 0.202i)21-s + 0.459i·23-s + 0.200i·25-s + (−0.763 − 0.763i)27-s + (0.232 − 0.232i)29-s − 0.366·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5857066871\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5857066871\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (0.925 - 0.925i)T - 3iT^{2} \) |
| 11 | \( 1 + (3.01 + 3.01i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.350 - 0.350i)T - 13iT^{2} \) |
| 17 | \( 1 - 5.90T + 17T^{2} \) |
| 19 | \( 1 + (0.223 - 0.223i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.20iT - 23T^{2} \) |
| 29 | \( 1 + (-1.25 + 1.25i)T - 29iT^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 + (1.71 + 1.71i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.96iT - 41T^{2} \) |
| 43 | \( 1 + (-5.64 - 5.64i)T + 43iT^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + (1.26 + 1.26i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.11 - 3.11i)T + 59iT^{2} \) |
| 61 | \( 1 + (-2.39 + 2.39i)T - 61iT^{2} \) |
| 67 | \( 1 + (4.05 - 4.05i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.47iT - 71T^{2} \) |
| 73 | \( 1 - 6.05iT - 73T^{2} \) |
| 79 | \( 1 - 1.37T + 79T^{2} \) |
| 83 | \( 1 + (8.19 - 8.19i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.38iT - 89T^{2} \) |
| 97 | \( 1 + 6.87T + 97T^{2} \) |
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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
−9.594942767171601652857527573511, −8.219417830733864204771704660311, −8.037606372579835147040842485249, −7.08415255606742601224253756413, −5.88049644160191162778878454433, −5.38538857888108225246523888608, −4.61282336318663375426717638318, −3.68463694911578939902585390838, −2.75047897222863522694871439106, −1.17888730359093922667786516675,
0.23918601949048288143802859119, 1.68260368523195507185669993761, 2.83066372531429750415280308743, 3.75195428231035319174957643221, 4.95779128895401199010941570848, 5.60461432655545589153015294440, 6.45404682956770838758132651646, 7.26575156280171316400154022524, 7.73820238651105024299079606237, 8.683647851861107292558434897856