| L(s) = 1 | + (1.37 + 0.321i)2-s + (−0.194 + 0.0191i)3-s + (1.79 + 0.884i)4-s + (−0.448 + 0.838i)5-s + (−0.274 − 0.0361i)6-s + (0.394 − 1.98i)7-s + (2.18 + 1.79i)8-s + (−2.90 + 0.577i)9-s + (−0.886 + 1.01i)10-s + (1.96 − 1.61i)11-s + (−0.366 − 0.137i)12-s + (−3.43 + 1.83i)13-s + (1.18 − 2.60i)14-s + (0.0712 − 0.172i)15-s + (2.43 + 3.17i)16-s + (−2.58 − 6.25i)17-s + ⋯ |
| L(s) = 1 | + (0.973 + 0.227i)2-s + (−0.112 + 0.0110i)3-s + (0.896 + 0.442i)4-s + (−0.200 + 0.375i)5-s + (−0.112 − 0.0147i)6-s + (0.149 − 0.749i)7-s + (0.773 + 0.634i)8-s + (−0.968 + 0.192i)9-s + (−0.280 + 0.319i)10-s + (0.591 − 0.485i)11-s + (−0.105 − 0.0398i)12-s + (−0.952 + 0.509i)13-s + (0.315 − 0.696i)14-s + (0.0184 − 0.0444i)15-s + (0.608 + 0.793i)16-s + (−0.627 − 1.51i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.62901 + 0.338223i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.62901 + 0.338223i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.37 - 0.321i)T \) |
| good | 3 | \( 1 + (0.194 - 0.0191i)T + (2.94 - 0.585i)T^{2} \) |
| 5 | \( 1 + (0.448 - 0.838i)T + (-2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-0.394 + 1.98i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.96 + 1.61i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (3.43 - 1.83i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (2.58 + 6.25i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-1.58 - 0.481i)T + (15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (4.28 + 2.86i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (3.65 - 4.45i)T + (-5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (1.49 + 1.49i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.443 - 1.46i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-1.49 + 2.23i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-10.6 - 1.05i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (1.16 - 0.483i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-5.69 - 6.93i)T + (-10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (1.42 + 0.761i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.332 - 3.37i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (0.689 + 6.99i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (7.67 + 1.52i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.201 - 1.01i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-13.7 - 5.68i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (4.86 - 16.0i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-3.43 + 2.29i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (2.61 + 2.61i)T + 97iT^{2} \) |
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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
−13.88652973577278203056725529470, −12.34734421677500706895817564684, −11.44858832526688242639254044723, −10.81356096432277234563760535091, −9.157673383030713216975799955780, −7.61723990626742097614671714435, −6.81968061437001885752769531783, −5.46502652224407135104389729903, −4.20437919970744213361785557843, −2.74807047629660629955399762809,
2.31774303233298479256069002590, 4.00648795899222448798296577920, 5.35599550690896296346651729793, 6.26876987761105405330152098634, 7.80664412898919681221695353818, 9.113892659434563214486975913960, 10.43516904324412324326654474066, 11.66857183307663438416410174160, 12.20162109319998872676656227536, 13.09320794422319658183193402657
