L(s) = 1 | + (−0.923 − 0.382i)3-s + (−0.541 + 0.541i)7-s + (0.707 + 0.707i)9-s + (0.707 − 1.70i)13-s + (−0.541 + 1.30i)19-s + (0.707 − 0.292i)21-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + 1.84·31-s + (−0.292 − 0.707i)37-s + (−1.30 + 1.30i)39-s + (1.30 − 0.541i)43-s + 0.414i·49-s + (1 − 0.999i)57-s + (1.70 + 0.707i)61-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)3-s + (−0.541 + 0.541i)7-s + (0.707 + 0.707i)9-s + (0.707 − 1.70i)13-s + (−0.541 + 1.30i)19-s + (0.707 − 0.292i)21-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + 1.84·31-s + (−0.292 − 0.707i)37-s + (−1.30 + 1.30i)39-s + (1.30 − 0.541i)43-s + 0.414i·49-s + (1 − 0.999i)57-s + (1.70 + 0.707i)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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8536389875\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8536389875\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.923 + 0.382i)T \) |
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 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 - 1.84T + T^{2} \) |
| 37 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 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
−8.724261737488872070689818382022, −8.025822661303036974296367804150, −7.40138794713837298644993141337, −6.33264623409890923780766734535, −5.86089703439046602277322043338, −5.37028843269634430243136029044, −4.20300187611851224233977691335, −3.28026118524072269078701162743, −2.17579258925687479251584308114, −0.881784627542399476360778456216,
0.873965147225082793822240325414, 2.27845352128515187543827568310, 3.62959717849181295420610447950, 4.33060080985551889237794741307, 4.88159478391971502484447493782, 6.13267034100045981473304136532, 6.55003676399062756695865688705, 7.06433045116298784324228369499, 8.249203599071522926652174727958, 9.073746937941166504584715680190