L(s) = 1 | + (0.236 + 0.409i)2-s + (−0.544 + 1.64i)3-s + (0.888 − 1.53i)4-s + (−0.802 + 0.165i)6-s + (1.28 + 2.21i)7-s + 1.78·8-s + (−2.40 − 1.79i)9-s + (3.08 + 5.34i)11-s + (2.04 + 2.29i)12-s + (1.06 − 1.84i)13-s + (−0.606 + 1.05i)14-s + (−1.35 − 2.34i)16-s − 3.16·17-s + (0.164 − 1.41i)18-s + 0.356·19-s + ⋯ |
L(s) = 1 | + (0.167 + 0.289i)2-s + (−0.314 + 0.949i)3-s + (0.444 − 0.769i)4-s + (−0.327 + 0.0676i)6-s + (0.484 + 0.838i)7-s + 0.631·8-s + (−0.802 − 0.597i)9-s + (0.929 + 1.61i)11-s + (0.590 + 0.663i)12-s + (0.295 − 0.512i)13-s + (−0.162 + 0.280i)14-s + (−0.338 − 0.585i)16-s − 0.768·17-s + (0.0388 − 0.332i)18-s + 0.0817·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19119 + 0.734646i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19119 + 0.734646i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.544 - 1.64i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.236 - 0.409i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-1.28 - 2.21i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.08 - 5.34i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.06 + 1.84i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 0.356T + 19T^{2} \) |
| 23 | \( 1 + (2.10 - 3.64i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.843 + 1.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.12 + 7.15i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.63T + 37T^{2} \) |
| 41 | \( 1 + (-1.36 + 2.36i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.83 + 6.64i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.71 + 9.89i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 9.43T + 53T^{2} \) |
| 59 | \( 1 + (5.10 - 8.84i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.00549 - 0.00952i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.491 + 0.851i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.43T + 71T^{2} \) |
| 73 | \( 1 + 6.61T + 73T^{2} \) |
| 79 | \( 1 + (-4.73 - 8.20i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.20 + 9.02i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 + (-3.60 - 6.24i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.96859784523715949183243022945, −11.56611307708050552155590568974, −10.35876654982223715858532819711, −9.688249577526710065287895337316, −8.699520464506919506688219877134, −7.16538610275875735733785477061, −6.04130653446336971978138511785, −5.16164973794139773323480894149, −4.14731925779046097513400458975, −2.07983919317509419131465660993,
1.41786713635791845432794324548, 3.12286882801594201057921738755, 4.44600132347031081595716436231, 6.25118614478839680652129305564, 6.91467346983301032970580820558, 8.083182704106452331490750245733, 8.745094118672273827953125705220, 10.72456557010999786841229299139, 11.27861375595164819282426845639, 11.94793052285194888291518559433