L(s) = 1 | + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.292 + 0.707i)11-s + (1 − i)23-s + (0.707 + 0.707i)25-s + (0.292 + 0.707i)29-s + (1.70 + 0.707i)37-s + (−0.707 + 1.70i)43-s + 1.00i·49-s + (−0.707 + 1.70i)53-s + 1.00·63-s + (0.707 + 1.70i)67-s + (0.707 − 0.292i)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 + (−0.292 + 0.707i)11-s + (1 − i)23-s + (0.707 + 0.707i)25-s + (0.292 + 0.707i)29-s + (1.70 + 0.707i)37-s + (−0.707 + 1.70i)43-s + 1.00i·49-s + (−0.707 + 1.70i)53-s + 1.00·63-s + (0.707 + 1.70i)67-s + (0.707 − 0.292i)77-s − 1.41i·79-s − 1.00i·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9708000650\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9708000650\) |
\(L(1)\) |
|
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 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 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 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 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
−8.864505864018625686613756459805, −8.045611378025312251328684597954, −7.36554843045136771436257716824, −6.68672465904155037285660784957, −5.94241032697833497107627595968, −4.88521595745778831048057574183, −4.45390107142037661639832817145, −3.14023172761004517891725750238, −2.65257611088873913911994785542, −1.20205910977634357760390577406,
0.62417873863026576592324344489, 2.30252273172149934569813484626, 3.10650199473350869445964997173, 3.72685142504968417407521490912, 5.00603185741370927174777646266, 5.70175724244050436504559984286, 6.32365705017853690458337977366, 6.98569125567371733398520373644, 8.092202497929530893230421368081, 8.616508497985280282738783644406