L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−2.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s + 13-s + 0.999·15-s + (2 − 3.46i)17-s + (−0.5 − 0.866i)19-s + (2 − 1.73i)21-s + (2 + 3.46i)23-s + (−0.499 + 0.866i)25-s + 0.999·27-s + (−2.5 + 4.33i)31-s + (−0.999 − 1.73i)33-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.944 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + 0.277·13-s + 0.258·15-s + (0.485 − 0.840i)17-s + (−0.114 − 0.198i)19-s + (0.436 − 0.377i)21-s + (0.417 + 0.722i)23-s + (−0.0999 + 0.173i)25-s + 0.192·27-s + (−0.449 + 0.777i)31-s + (−0.174 − 0.301i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.028529678\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.028529678\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.5 + 7.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 18T + 83T^{2} \) |
| 89 | \( 1 + (-2 - 3.46i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.372960511496372624888970697284, −9.068474821579911004764460608243, −7.79137326155969896130883151046, −7.16411723315561478427875274431, −6.22140240530667056451621708955, −5.34989908838587901097000897222, −4.54253484675341421123515623297, −3.63010181154341995121709656309, −2.73511992278929685097364155776, −0.986972068742934260807236016782,
0.51004141795929321380113246737, 2.13999297252676775047634171269, 3.16106563705712369170331132160, 4.00033967287986069340874418936, 5.35945929500864102916580883513, 6.12311683800308626604697245279, 6.62954887090789796740107830443, 7.64386792658997494728107268695, 8.288658907916360294317972083189, 9.203159774340059391327469389048