L(s) = 1 | + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.707 + 1.70i)11-s + (1 − i)23-s + (0.707 + 0.707i)25-s + (−0.707 − 1.70i)29-s + (0.707 + 0.292i)37-s + (0.292 − 0.707i)43-s + 1.00i·49-s + (−0.292 + 0.707i)53-s − 1.00·63-s + (−0.292 − 0.707i)67-s + (−1.70 + 0.707i)77-s − 1.41i·79-s − 1.00i·81-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.707 + 1.70i)11-s + (1 − i)23-s + (0.707 + 0.707i)25-s + (−0.707 − 1.70i)29-s + (0.707 + 0.292i)37-s + (0.292 − 0.707i)43-s + 1.00i·49-s + (−0.292 + 0.707i)53-s − 1.00·63-s + (−0.292 − 0.707i)67-s + (−1.70 + 0.707i)77-s − 1.41i·79-s − 1.00i·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9684709581\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9684709581\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.54301364278793173580832314525, −9.570677947728051515882716116117, −8.755953910041500744601541054870, −7.87941309732915751147963891809, −7.26823706403039517249601387275, −5.99864746304311802393021972678, −5.05266164470524463309420157174, −4.51445647198220240170724483400, −2.75027351849633254106181210761, −2.00195672710065480054624641733,
1.02917740701153812391221961772, 2.90007727478073064167155416647, 3.67321395568854535224913615701, 5.03038666312973927882021152941, 5.74965680322947126483517689282, 6.77820815604410505503678097931, 7.75667050804081421545669569929, 8.543719113023377707721609270467, 9.161401084981648883275826620811, 10.37867037697018834474307957650