L(s) = 1 | + (0.5 − 0.866i)5-s + (1.32 − 2.29i)7-s + (1.82 + 3.15i)11-s + 2.64·13-s + (1.82 + 3.15i)17-s + (1.14 − 1.98i)19-s + (−1.82 + 3.15i)23-s + (−0.499 − 0.866i)25-s − 2.35·29-s + (−3.14 − 5.44i)31-s + (−1.32 − 2.29i)35-s + (2.32 − 4.02i)37-s + 10.9·41-s + 9.93·43-s + (3 − 5.19i)47-s + ⋯ |
L(s) = 1 | + (0.223 − 0.387i)5-s + (0.499 − 0.866i)7-s + (0.549 + 0.951i)11-s + 0.733·13-s + (0.442 + 0.765i)17-s + (0.262 − 0.455i)19-s + (−0.380 + 0.658i)23-s + (−0.0999 − 0.173i)25-s − 0.437·29-s + (−0.564 − 0.978i)31-s + (−0.223 − 0.387i)35-s + (0.381 − 0.661i)37-s + 1.70·41-s + 1.51·43-s + (0.437 − 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.968458277\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.968458277\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-1.32 + 2.29i)T \) |
good | 11 | \( 1 + (-1.82 - 3.15i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 + (-1.82 - 3.15i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.14 + 1.98i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.82 - 3.15i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.35T + 29T^{2} \) |
| 31 | \( 1 + (3.14 + 5.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.32 + 4.02i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 9.93T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.64 + 6.31i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.46 + 4.27i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.32 - 4.02i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.35T + 71T^{2} \) |
| 73 | \( 1 + (-6.61 - 11.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.14 + 7.18i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.93T + 83T^{2} \) |
| 89 | \( 1 + (-6.11 + 10.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−9.556286422419796621092715305505, −8.960347558840496830936748774436, −7.80563344608795027497130913434, −7.38941703503139042160467916727, −6.25347578913483579641513818144, −5.44019881541301818436715284472, −4.28497816259393512866648339923, −3.79662224238735448461526805623, −2.09667361890457538878958205230, −1.05728629360491220302824439036,
1.22800476100533845974334487086, 2.57417255265591881432418633395, 3.48873397837469220704955005331, 4.65858099046749551853078638757, 5.85417019853883451977584433147, 6.10183484186941932344513029080, 7.38122139972552554006121466878, 8.170679536900099714137701244292, 9.020385647097935025823120173543, 9.525469091439700638519961138507