L(s) = 1 | + (−1.15 + 0.812i)2-s + (0.678 − 1.88i)4-s + (−2.39 − 2.39i)5-s + 2.42i·7-s + (0.744 + 2.72i)8-s + (4.71 + 0.824i)10-s + (−0.803 − 0.803i)11-s + (0.643 − 0.643i)13-s + (−1.97 − 2.81i)14-s + (−3.07 − 2.55i)16-s − 0.717·17-s + (−3.76 + 3.76i)19-s + (−6.12 + 2.87i)20-s + (1.58 + 0.276i)22-s + 7.35i·23-s + ⋯ |
L(s) = 1 | + (−0.818 + 0.574i)2-s + (0.339 − 0.940i)4-s + (−1.07 − 1.07i)5-s + 0.918i·7-s + (0.263 + 0.964i)8-s + (1.49 + 0.260i)10-s + (−0.242 − 0.242i)11-s + (0.178 − 0.178i)13-s + (−0.527 − 0.751i)14-s + (−0.769 − 0.638i)16-s − 0.174·17-s + (−0.864 + 0.864i)19-s + (−1.37 + 0.643i)20-s + (0.337 + 0.0589i)22-s + 1.53i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0310325 + 0.205696i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0310325 + 0.205696i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.15 - 0.812i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.39 + 2.39i)T + 5iT^{2} \) |
| 7 | \( 1 - 2.42iT - 7T^{2} \) |
| 11 | \( 1 + (0.803 + 0.803i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.643 + 0.643i)T - 13iT^{2} \) |
| 17 | \( 1 + 0.717T + 17T^{2} \) |
| 19 | \( 1 + (3.76 - 3.76i)T - 19iT^{2} \) |
| 23 | \( 1 - 7.35iT - 23T^{2} \) |
| 29 | \( 1 + (6.75 - 6.75i)T - 29iT^{2} \) |
| 31 | \( 1 + 6.69T + 31T^{2} \) |
| 37 | \( 1 + (7.02 + 7.02i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.53iT - 41T^{2} \) |
| 43 | \( 1 + (-1.31 - 1.31i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.02T + 47T^{2} \) |
| 53 | \( 1 + (6.85 + 6.85i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.81 - 4.81i)T + 59iT^{2} \) |
| 61 | \( 1 + (8.63 - 8.63i)T - 61iT^{2} \) |
| 67 | \( 1 + (-11.0 + 11.0i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.42iT - 71T^{2} \) |
| 73 | \( 1 - 14.0iT - 73T^{2} \) |
| 79 | \( 1 + 5.60T + 79T^{2} \) |
| 83 | \( 1 + (-6.14 + 6.14i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.44iT - 89T^{2} \) |
| 97 | \( 1 + 7.77T + 97T^{2} \) |
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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
−11.43915675169458033815466365908, −10.67703280756539688436146592110, −9.292830668984817681910199941245, −8.828990366878285445576836507951, −8.021767207274008311855032304912, −7.26799119218699490742768751206, −5.79045672673561355407377406239, −5.18119423227341639786248534029, −3.71704732308834093815785411818, −1.72148943624222253225272537398,
0.16741475413730415675664326726, 2.34527171688737838734055096867, 3.59894100243178529759751097675, 4.36389459100296423043776534039, 6.58296985310416486255447901094, 7.19924039613658723025269318319, 7.951548874248590532213775210855, 8.925131353559016900237852226781, 10.13005118516821273492667772526, 10.86641325866149562175904340050