| L(s) = 1 | + (−1.03 + 0.959i)2-s + (0.130 + 1.72i)3-s + (0.158 − 1.99i)4-s + (0.588 − 0.214i)5-s + (−1.79 − 1.66i)6-s + (−4.57 + 0.806i)7-s + (1.74 + 2.22i)8-s + (−2.96 + 0.450i)9-s + (−0.405 + 0.787i)10-s + (−1.37 + 3.76i)11-s + (3.46 + 0.0137i)12-s + (−0.583 + 0.695i)13-s + (3.97 − 5.22i)14-s + (0.446 + 0.988i)15-s + (−3.94 − 0.632i)16-s + (2.52 + 1.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.734 + 0.678i)2-s + (0.0752 + 0.997i)3-s + (0.0792 − 0.996i)4-s + (0.263 − 0.0958i)5-s + (−0.731 − 0.681i)6-s + (−1.72 + 0.304i)7-s + (0.618 + 0.786i)8-s + (−0.988 + 0.150i)9-s + (−0.128 + 0.249i)10-s + (−0.413 + 1.13i)11-s + (0.999 + 0.00396i)12-s + (−0.161 + 0.192i)13-s + (1.06 − 1.39i)14-s + (0.115 + 0.255i)15-s + (−0.987 − 0.158i)16-s + (0.613 + 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0195200 - 0.500159i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0195200 - 0.500159i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.03 - 0.959i)T \) |
| 3 | \( 1 + (-0.130 - 1.72i)T \) |
| good | 5 | \( 1 + (-0.588 + 0.214i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (4.57 - 0.806i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (1.37 - 3.76i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.583 - 0.695i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.52 - 1.45i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.50 + 4.34i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.819 - 4.64i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-6.21 + 5.21i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.29 - 0.756i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.72 - 2.15i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.41 - 6.44i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.48 - 1.99i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.38 - 7.85i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 3.06T + 53T^{2} \) |
| 59 | \( 1 + (0.537 + 1.47i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (3.32 - 0.586i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-9.02 - 7.57i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (3.19 - 5.53i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.18 + 3.78i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.52 + 6.58i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.42 - 7.65i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (1.88 - 1.08i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.312 + 0.113i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.93769425776364831461622388782, −11.63584464634320628029342149725, −10.27195639680579338761614272303, −9.743768427979764666196892563541, −9.248644089734494265818131264193, −7.953881289021808894017257107358, −6.63384040762298804112973400204, −5.75846241933607237675626910759, −4.49495381588219453522740314295, −2.72716598406182984528772804874,
0.50215091853556760445383910434, 2.57959819124783014783513469206, 3.51718226878194847805040422778, 6.00023523043538564373378308973, 6.79471040687398475013772725532, 7.975989781614895762400221312003, 8.830507621833725077119237342057, 10.02289203789145366529002732488, 10.64323882283895593451705809164, 12.08779022645329340583473552251
