L(s) = 1 | + (0.707 − 0.707i)3-s + (−2.23 + 0.146i)5-s + (2.93 − 2.93i)7-s − 1.00i·9-s + 6.31i·11-s + (0.106 − 0.106i)13-s + (−1.47 + 1.68i)15-s + (−2.74 + 2.74i)17-s + 7.24·19-s − 4.14i·21-s + (4.53 + 1.57i)23-s + (4.95 − 0.654i)25-s + (−0.707 − 0.707i)27-s − 9.94i·29-s + 6.76·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.997 + 0.0655i)5-s + (1.10 − 1.10i)7-s − 0.333i·9-s + 1.90i·11-s + (0.0295 − 0.0295i)13-s + (−0.380 + 0.434i)15-s + (−0.666 + 0.666i)17-s + 1.66·19-s − 0.905i·21-s + (0.944 + 0.327i)23-s + (0.991 − 0.130i)25-s + (−0.136 − 0.136i)27-s − 1.84i·29-s + 1.21·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.892486229\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.892486229\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.23 - 0.146i)T \) |
| 23 | \( 1 + (-4.53 - 1.57i)T \) |
good | 7 | \( 1 + (-2.93 + 2.93i)T - 7iT^{2} \) |
| 11 | \( 1 - 6.31iT - 11T^{2} \) |
| 13 | \( 1 + (-0.106 + 0.106i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.74 - 2.74i)T - 17iT^{2} \) |
| 19 | \( 1 - 7.24T + 19T^{2} \) |
| 29 | \( 1 + 9.94iT - 29T^{2} \) |
| 31 | \( 1 - 6.76T + 31T^{2} \) |
| 37 | \( 1 + (1.65 - 1.65i)T - 37iT^{2} \) |
| 41 | \( 1 - 8.29T + 41T^{2} \) |
| 43 | \( 1 + (5.22 + 5.22i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.00 + 3.00i)T + 47iT^{2} \) |
| 53 | \( 1 + (8.34 + 8.34i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.05iT - 59T^{2} \) |
| 61 | \( 1 + 1.36iT - 61T^{2} \) |
| 67 | \( 1 + (4.30 - 4.30i)T - 67iT^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + (5.69 - 5.69i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 + (-4.73 - 4.73i)T + 83iT^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + (-3.23 + 3.23i)T - 97iT^{2} \) |
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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
−9.593421940074517177053960893574, −8.424292617176447802876360150184, −7.71121041715577285276926503195, −7.36885206681959874378820614480, −6.61695184868743811207151897523, −4.94192732624595880485529706351, −4.43571226256652684793098689868, −3.55228923766432530612235131471, −2.15445213139062810575499014822, −1.00721913506444333475353437949,
1.08117091681771733483664105340, 2.86062007259225195784532814221, 3.32114114206508344282047599708, 4.76350464238126601501393548223, 5.16284936665971491078140157032, 6.29299181154327174767168459402, 7.54001566139146637558953948512, 8.138576162528427380059270121939, 8.908566497616147895339867624183, 9.168165137929518631438779524547