L(s) = 1 | − 1.73i·3-s + (−2.13 + 0.656i)5-s + 2.27·11-s + 6.09i·13-s + (1.13 + 3.70i)15-s + 4.77i·17-s − 4.27·19-s + 0.894i·23-s + (4.13 − 2.80i)25-s − 5.19i·27-s + 3.27·29-s + 4.27·31-s − 3.94i·33-s + 5.61i·37-s + 10.5·39-s + ⋯ |
L(s) = 1 | − 0.999i·3-s + (−0.955 + 0.293i)5-s + 0.685·11-s + 1.68i·13-s + (0.293 + 0.955i)15-s + 1.15i·17-s − 0.980·19-s + 0.186i·23-s + (0.827 − 0.561i)25-s − 1.00i·27-s + 0.608·29-s + 0.767·31-s − 0.685i·33-s + 0.923i·37-s + 1.68·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27679 + 0.191718i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27679 + 0.191718i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.13 - 0.656i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 1.73iT - 3T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 - 6.09iT - 13T^{2} \) |
| 17 | \( 1 - 4.77iT - 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 - 0.894iT - 23T^{2} \) |
| 29 | \( 1 - 3.27T + 29T^{2} \) |
| 31 | \( 1 - 4.27T + 31T^{2} \) |
| 37 | \( 1 - 5.61iT - 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 6.50iT - 43T^{2} \) |
| 47 | \( 1 - 2.15iT - 47T^{2} \) |
| 53 | \( 1 - 7.40iT - 53T^{2} \) |
| 59 | \( 1 - 4.27T + 59T^{2} \) |
| 61 | \( 1 + 1.54T + 61T^{2} \) |
| 67 | \( 1 - 13.9iT - 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 2.15iT - 73T^{2} \) |
| 79 | \( 1 - 0.274T + 79T^{2} \) |
| 83 | \( 1 + 5.67iT - 83T^{2} \) |
| 89 | \( 1 + 7T + 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−10.11209279550124299917181499929, −8.955004722327041218060134857513, −8.298072741620398516193549685264, −7.42224451975405123933934010116, −6.65846745887568499781753674940, −6.23353756148679992909265695095, −4.41499023462223493689252778271, −3.96066028588152362994559450636, −2.39015963228159148563648643042, −1.25118665028775341746429106589,
0.70309097798995950529974344548, 2.83701511865367613755596623688, 3.81604207150010575292110028939, 4.56363595166262547617027223139, 5.34291805562788490287603599609, 6.58796711506275880962324399771, 7.61376913391591051322402423651, 8.336940311184180760971633343218, 9.217210664525578706332734492160, 9.942954240389142451806043746003