L(s) = 1 | + (−2.44 − 2.44i)5-s − 1.41i·7-s + (3.46 + 3.46i)11-s + (−1 + i)13-s − 4.89·17-s + (−4.24 + 4.24i)19-s − 6.92i·23-s + 6.99i·25-s + (−2.44 + 2.44i)29-s + 1.41·31-s + (−3.46 + 3.46i)35-s + (7 + 7i)37-s − 4.89i·41-s + (4.24 + 4.24i)43-s − 6.92·47-s + ⋯ |
L(s) = 1 | + (−1.09 − 1.09i)5-s − 0.534i·7-s + (1.04 + 1.04i)11-s + (−0.277 + 0.277i)13-s − 1.18·17-s + (−0.973 + 0.973i)19-s − 1.44i·23-s + 1.39i·25-s + (−0.454 + 0.454i)29-s + 0.254·31-s + (−0.585 + 0.585i)35-s + (1.15 + 1.15i)37-s − 0.765i·41-s + (0.646 + 0.646i)43-s − 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7991251589\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7991251589\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.44 + 2.44i)T + 5iT^{2} \) |
| 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 + (-3.46 - 3.46i)T + 11iT^{2} \) |
| 13 | \( 1 + (1 - i)T - 13iT^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + (4.24 - 4.24i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 + (2.44 - 2.44i)T - 29iT^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + (-7 - 7i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.89iT - 41T^{2} \) |
| 43 | \( 1 + (-4.24 - 4.24i)T + 43iT^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + (-2.44 - 2.44i)T + 53iT^{2} \) |
| 59 | \( 1 + (6.92 + 6.92i)T + 59iT^{2} \) |
| 61 | \( 1 + (5 - 5i)T - 61iT^{2} \) |
| 67 | \( 1 + (5.65 - 5.65i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + (-10.3 + 10.3i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.79iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.972136272548539627741720649665, −8.458415472597212389933100491006, −7.67223015975509533082918527588, −6.88273181996290599081551954950, −6.20450712757455756666085283107, −4.69100237004295614709207579045, −4.43009952245942420694844268639, −3.80449850334678184012602791907, −2.21616301020542578668076544150, −1.05723979810389542253047699728,
0.31790769441746445381781981961, 2.16315755230333119864710357551, 3.13916853697289742398110358409, 3.84330315839403386843539097244, 4.69659845624383823503642895345, 6.02827584217919539976704400190, 6.46759489996860906231984871599, 7.37939452313606726865420839489, 7.969858752082834491940313884988, 8.980471961139831406852265606932