| L(s) = 1 | + (−1.38 − 0.268i)2-s + (−1.38 − 1.04i)3-s + (1.85 + 0.746i)4-s + (−3.21 − 1.16i)5-s + (1.63 + 1.82i)6-s + (0.199 + 0.0351i)7-s + (−2.37 − 1.53i)8-s + (0.811 + 2.88i)9-s + (4.14 + 2.48i)10-s + (1.37 + 3.78i)11-s + (−1.78 − 2.97i)12-s + (−1.09 − 1.30i)13-s + (−0.267 − 0.102i)14-s + (3.21 + 4.97i)15-s + (2.88 + 2.77i)16-s + (1.57 − 0.908i)17-s + ⋯ |
| L(s) = 1 | + (−0.981 − 0.190i)2-s + (−0.796 − 0.603i)3-s + (0.927 + 0.373i)4-s + (−1.43 − 0.523i)5-s + (0.667 + 0.744i)6-s + (0.0752 + 0.0132i)7-s + (−0.839 − 0.542i)8-s + (0.270 + 0.962i)9-s + (1.31 + 0.786i)10-s + (0.415 + 1.14i)11-s + (−0.513 − 0.857i)12-s + (−0.304 − 0.362i)13-s + (−0.0714 − 0.0273i)14-s + (0.829 + 1.28i)15-s + (0.721 + 0.692i)16-s + (0.381 − 0.220i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.175501 + 0.153429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.175501 + 0.153429i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 + 0.268i)T \) |
| 3 | \( 1 + (1.38 + 1.04i)T \) |
| good | 5 | \( 1 + (3.21 + 1.16i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.199 - 0.0351i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.37 - 3.78i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.09 + 1.30i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.57 + 0.908i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.24 - 5.61i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.37 - 7.80i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.08 - 2.58i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (7.51 - 1.32i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.53 - 0.885i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.61 + 7.88i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.47 + 1.26i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.523 - 2.97i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 9.46T + 53T^{2} \) |
| 59 | \( 1 + (-1.23 + 3.38i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-8.13 - 1.43i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.26 - 1.06i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.24 - 3.89i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.05 - 7.01i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.380 + 0.453i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (8.02 - 9.56i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-10.7 - 6.18i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.24 - 3.00i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.28629613844215954174760987348, −11.66157912142531152667252333808, −10.74763437727988560176986834321, −9.673558551067765788246195720525, −8.338532530615833784988945046419, −7.58241366528468355871033944087, −6.90516280847524446030909092365, −5.30537178375580627275600792030, −3.78134016489590344322033214725, −1.56614769626547355163546920393,
0.30372336005310850224830561492, 3.21899191279487672641567126196, 4.60328803971469419318522658760, 6.24241277810075702564762867990, 6.98874254041595741254515657824, 8.226849214978769924645232766349, 9.068432080301417206604582671841, 10.36919884044640303399148957729, 11.18682476354523308754607070166, 11.50318949475432616590310375528
