L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.707 − 1.22i)3-s + (−0.499 − 0.866i)4-s + (1.41 − 2.44i)5-s + 1.41·6-s + 0.999·8-s + (0.500 − 0.866i)9-s + (1.41 + 2.44i)10-s + (1 + 1.73i)11-s + (−0.707 + 1.22i)12-s − 4·15-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s + (0.499 + 0.866i)18-s + (−3.53 + 6.12i)19-s − 2.82·20-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.408 − 0.707i)3-s + (−0.249 − 0.433i)4-s + (0.632 − 1.09i)5-s + 0.577·6-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.447 + 0.774i)10-s + (0.301 + 0.522i)11-s + (−0.204 + 0.353i)12-s − 1.03·15-s + (−0.125 + 0.216i)16-s + (−0.171 − 0.297i)17-s + (0.117 + 0.204i)18-s + (−0.811 + 1.40i)19-s − 0.632·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.778377 - 0.239171i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.778377 - 0.239171i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.707 + 1.22i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.41 + 2.44i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.53 - 6.12i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (-4.24 - 7.34i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.707 + 1.22i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.41 + 2.44i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (0.707 + 1.22i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + (3.53 - 6.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−13.76193809812566124817678659413, −12.68605884014301529294020347923, −12.12565423005745550882922386484, −10.36797756766802892753769301082, −9.287691601674165385886858031133, −8.299776803105340755450334301620, −6.89861159000452862380207722778, −5.94916008639523159694127953550, −4.61592896308588504442376428583, −1.40536345694091675296508686664,
2.55959814979660381660161526943, 4.24208572076520311235406283664, 5.89273232430019475483528776608, 7.26568928480256498708708581767, 8.934524743459123306035708801029, 9.987839736584902093994878144698, 10.84562124592142148262248640429, 11.36040525061874285112173456885, 13.00642752127750995048478279999, 13.91972604691128846033682844961