L(s) = 1 | + (−0.766 + 0.642i)5-s + (0.5 − 0.866i)7-s + (−0.939 + 0.342i)9-s + (0.5 + 0.866i)11-s + (0.939 + 0.342i)17-s + (1.53 + 1.28i)23-s + (0.173 + 0.984i)35-s + (−0.766 + 0.642i)43-s + (0.5 − 0.866i)45-s + (0.939 − 0.342i)47-s + (−0.939 − 0.342i)55-s + (−0.766 − 0.642i)61-s + (−0.173 + 0.984i)63-s + (−0.173 − 0.984i)73-s + 0.999·77-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)5-s + (0.5 − 0.866i)7-s + (−0.939 + 0.342i)9-s + (0.5 + 0.866i)11-s + (0.939 + 0.342i)17-s + (1.53 + 1.28i)23-s + (0.173 + 0.984i)35-s + (−0.766 + 0.642i)43-s + (0.5 − 0.866i)45-s + (0.939 − 0.342i)47-s + (−0.939 − 0.342i)55-s + (−0.766 − 0.642i)61-s + (−0.173 + 0.984i)63-s + (−0.173 − 0.984i)73-s + 0.999·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9661500862\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9661500862\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
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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
−9.886940712231416103057073376773, −9.015351979790773241431444824681, −7.985316922466200420419025459704, −7.45966791389632421992738826988, −6.84605813483333681213953026707, −5.64054992015472729597704667910, −4.74564587462473195367746254399, −3.75130509486854263393211824391, −2.99790497160659598509919664440, −1.46007341096216204260244962391,
0.903241290934177371385519360749, 2.63241949588713765969114059666, 3.50713896686204169279568213297, 4.65806610128235004011945769301, 5.44707631180684654849215423146, 6.19314474527644477021500015214, 7.29665723921618973831742696989, 8.396830109110692318965843025206, 8.612875827655572386000674657600, 9.278978413018265241485753811707