L(s) = 1 | + (−0.382 − 0.923i)3-s + (0.541 + 0.541i)7-s + (−0.707 + 0.707i)9-s + (−1.70 + 0.707i)13-s + (1.30 − 0.541i)19-s + (0.292 − 0.707i)21-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + 1.84·31-s + (0.707 + 0.292i)37-s + (1.30 + 1.30i)39-s + (0.541 − 1.30i)43-s − 0.414i·49-s + (−1 − 0.999i)57-s + (0.707 + 1.70i)61-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.923i)3-s + (0.541 + 0.541i)7-s + (−0.707 + 0.707i)9-s + (−1.70 + 0.707i)13-s + (1.30 − 0.541i)19-s + (0.292 − 0.707i)21-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + 1.84·31-s + (0.707 + 0.292i)37-s + (1.30 + 1.30i)39-s + (0.541 − 1.30i)43-s − 0.414i·49-s + (−1 − 0.999i)57-s + (0.707 + 1.70i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.082092875\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082092875\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.382 + 0.923i)T \) |
good | 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 - 1.84T + T^{2} \) |
| 37 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 79 | \( 1 - 1.84iT - T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - 1.41T + T^{2} \) |
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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
−8.746933828619469945438604439850, −8.027321915574541492814134226412, −7.14892463896216252344224774877, −6.92944794477735590906481380167, −5.72609713724053301203920392014, −5.15287329820091487437572754208, −4.46144734508467925407113644957, −2.86461933181926851371319345237, −2.29547769729968623158787986962, −1.09408949258608900793198075226,
0.874066202817851605154422954228, 2.58861889459835618237477364216, 3.34067078736886783112098270142, 4.63283929385568221225354052452, 4.74423814899469124330608870646, 5.73158662903428268892128084496, 6.56113903153948047984087735544, 7.63969383692973971213501642022, 7.987139064270024483336646742731, 9.102776013122784541717036832424