L(s) = 1 | + (1.15 − 1.15i)2-s − 1.98i·3-s − 0.678i·4-s + (2.02 + 2.02i)5-s + (−2.29 − 2.29i)6-s + (1.52 + 1.52i)8-s − 0.925·9-s + 4.69·10-s + (2.44 + 2.44i)11-s − 1.34·12-s + (2.29 − 2.78i)13-s + (4.02 − 4.02i)15-s + 4.89·16-s − 6.44·17-s + (−1.07 + 1.07i)18-s + (−2.14 − 2.14i)19-s + ⋯ |
L(s) = 1 | + (0.818 − 0.818i)2-s − 1.14i·3-s − 0.339i·4-s + (0.907 + 0.907i)5-s + (−0.936 − 0.936i)6-s + (0.540 + 0.540i)8-s − 0.308·9-s + 1.48·10-s + (0.737 + 0.737i)11-s − 0.387·12-s + (0.635 − 0.771i)13-s + (1.03 − 1.03i)15-s + 1.22·16-s − 1.56·17-s + (−0.252 + 0.252i)18-s + (−0.493 − 0.493i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.26464 - 1.63774i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.26464 - 1.63774i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.29 + 2.78i)T \) |
good | 2 | \( 1 + (-1.15 + 1.15i)T - 2iT^{2} \) |
| 3 | \( 1 + 1.98iT - 3T^{2} \) |
| 5 | \( 1 + (-2.02 - 2.02i)T + 5iT^{2} \) |
| 11 | \( 1 + (-2.44 - 2.44i)T + 11iT^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 + (2.14 + 2.14i)T + 19iT^{2} \) |
| 23 | \( 1 - 3.43iT - 23T^{2} \) |
| 29 | \( 1 + 4.40T + 29T^{2} \) |
| 31 | \( 1 + (-2.37 - 2.37i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.75 + 2.75i)T + 37iT^{2} \) |
| 41 | \( 1 + (3.03 + 3.03i)T + 41iT^{2} \) |
| 43 | \( 1 + 4.48iT - 43T^{2} \) |
| 47 | \( 1 + (4.03 - 4.03i)T - 47iT^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + (2.09 - 2.09i)T - 59iT^{2} \) |
| 61 | \( 1 - 3.50iT - 61T^{2} \) |
| 67 | \( 1 + (-5.15 + 5.15i)T - 67iT^{2} \) |
| 71 | \( 1 + (-8.31 + 8.31i)T - 71iT^{2} \) |
| 73 | \( 1 + (3.30 - 3.30i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.08T + 79T^{2} \) |
| 83 | \( 1 + (-2.01 - 2.01i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.50 + 3.50i)T - 89iT^{2} \) |
| 97 | \( 1 + (3.20 + 3.20i)T + 97iT^{2} \) |
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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
−10.78710474184255208061576899380, −9.799004310873031393221779406891, −8.650274138839750062214326588964, −7.48688814925177318339248629927, −6.72943453797866135116639116607, −6.01775100416550878812070283532, −4.71011437724796555850026053017, −3.50297849790485106231855997651, −2.31860642213577977780117733567, −1.70477550862374519642337764094,
1.62104532620399154595869489242, 3.73435328855909849951059071225, 4.46278400982487018684205621189, 5.12353312046401868719677865697, 6.19215884372722617138661294939, 6.63224714978683324531459802478, 8.337316209381537137746447268178, 9.116959736226850806728116197756, 9.704733944846932931801128467912, 10.67462071844693478557299670521