L(s) = 1 | + (1.24 + 0.663i)2-s + (0.707 + 0.707i)3-s + (1.12 + 1.65i)4-s + (1.67 − 1.67i)5-s + (0.414 + 1.35i)6-s − i·7-s + (0.301 + 2.81i)8-s + 1.00i·9-s + (3.21 − 0.984i)10-s + (0.497 − 0.497i)11-s + (−0.379 + 1.96i)12-s + (−3.02 − 3.02i)13-s + (0.663 − 1.24i)14-s + 2.37·15-s + (−1.48 + 3.71i)16-s − 6.75·17-s + ⋯ |
L(s) = 1 | + (0.883 + 0.468i)2-s + (0.408 + 0.408i)3-s + (0.560 + 0.828i)4-s + (0.751 − 0.751i)5-s + (0.169 + 0.552i)6-s − 0.377i·7-s + (0.106 + 0.994i)8-s + 0.333i·9-s + (1.01 − 0.311i)10-s + (0.149 − 0.149i)11-s + (−0.109 + 0.566i)12-s + (−0.837 − 0.837i)13-s + (0.177 − 0.333i)14-s + 0.613·15-s + (−0.372 + 0.928i)16-s − 1.63·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.38099 + 1.00216i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.38099 + 1.00216i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.24 - 0.663i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + (-1.67 + 1.67i)T - 5iT^{2} \) |
| 11 | \( 1 + (-0.497 + 0.497i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.02 + 3.02i)T + 13iT^{2} \) |
| 17 | \( 1 + 6.75T + 17T^{2} \) |
| 19 | \( 1 + (-2.78 - 2.78i)T + 19iT^{2} \) |
| 23 | \( 1 - 0.0553iT - 23T^{2} \) |
| 29 | \( 1 + (1.37 + 1.37i)T + 29iT^{2} \) |
| 31 | \( 1 + 0.851T + 31T^{2} \) |
| 37 | \( 1 + (2.22 - 2.22i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.21iT - 41T^{2} \) |
| 43 | \( 1 + (8.33 - 8.33i)T - 43iT^{2} \) |
| 47 | \( 1 - 8.67T + 47T^{2} \) |
| 53 | \( 1 + (-5.16 + 5.16i)T - 53iT^{2} \) |
| 59 | \( 1 + (-10.6 + 10.6i)T - 59iT^{2} \) |
| 61 | \( 1 + (1.30 + 1.30i)T + 61iT^{2} \) |
| 67 | \( 1 + (-0.0901 - 0.0901i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.54iT - 71T^{2} \) |
| 73 | \( 1 - 0.395iT - 73T^{2} \) |
| 79 | \( 1 - 9.36T + 79T^{2} \) |
| 83 | \( 1 + (-5.80 - 5.80i)T + 83iT^{2} \) |
| 89 | \( 1 + 11.7iT - 89T^{2} \) |
| 97 | \( 1 + 6.66T + 97T^{2} \) |
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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
−11.88412267484514019498913686453, −10.78553763015290855277561898864, −9.742481542541956428202356404713, −8.781106761704144224161709769496, −7.84123688756422776393120655221, −6.74077289254698647936164739051, −5.51081256099799202652257547181, −4.79536403576899711191190533741, −3.61380184329683693393194708719, −2.19054963234387882456663301459,
2.02506042371329113269890698322, 2.72682071521651496184353921414, 4.24559009709784953764660530168, 5.47392198667159779729203284964, 6.67919400460862452737135665316, 7.08463536091685417019634606657, 8.908678913905586358046184989833, 9.681462552200374415461008864002, 10.67436708951416751878492038396, 11.59896713893777892868236669105