L(s) = 1 | + (0.5 + 0.866i)5-s + (1.72 − 2.98i)7-s + (0.724 − 1.25i)11-s + (2.94 + 5.10i)13-s − 4.89·17-s + 4·19-s + (2.72 + 4.71i)23-s + (2 − 3.46i)25-s + (−0.0505 + 0.0874i)29-s + (1.27 + 2.20i)31-s + 3.44·35-s + 0.898·37-s + (−5.94 − 10.3i)41-s + (1.17 − 2.03i)43-s + (3.17 − 5.49i)47-s + ⋯ |
L(s) = 1 | + (0.223 + 0.387i)5-s + (0.651 − 1.12i)7-s + (0.218 − 0.378i)11-s + (0.818 + 1.41i)13-s − 1.18·17-s + 0.917·19-s + (0.568 + 0.984i)23-s + (0.400 − 0.692i)25-s + (−0.00937 + 0.0162i)29-s + (0.229 + 0.396i)31-s + 0.583·35-s + 0.147·37-s + (−0.929 − 1.60i)41-s + (0.179 − 0.310i)43-s + (0.463 − 0.801i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.065998170\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.065998170\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.72 + 2.98i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.724 + 1.25i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.94 - 5.10i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-2.72 - 4.71i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0505 - 0.0874i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.27 - 2.20i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.898T + 37T^{2} \) |
| 41 | \( 1 + (5.94 + 10.3i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.17 + 2.03i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.17 + 5.49i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 8.89T + 53T^{2} \) |
| 59 | \( 1 + (-7.17 - 12.4i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.94 - 6.84i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.17 + 10.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.79T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.275 - 0.476i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + (-1.94 + 3.37i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.043556190990965701859040056762, −8.785171700080501159457487586262, −7.51363841545800908509403922792, −6.98654862241536531562171246045, −6.27139523363594037707538634087, −5.14450345167735074413660997233, −4.21386504088834079915415819672, −3.55066214829680230130596966838, −2.13356210915103850722693014870, −1.04951655921322591573620054157,
1.07336079612621537608805950955, 2.29480020105080556231685410458, 3.23574429797695283510368643474, 4.58423533164180212757894368288, 5.24634318619143262840983201375, 5.97361148878903154157960562778, 6.88701627700647538406160476821, 8.014348212281432522867110797302, 8.535526474325837693306333513048, 9.192549985877580440630255961023