L(s) = 1 | − 0.0421i·2-s + (−0.440 − 0.440i)3-s + 1.99·4-s + (2.27 + 2.27i)5-s + (−0.0185 + 0.0185i)6-s + (−3.03 + 3.03i)7-s − 0.168i·8-s − 2.61i·9-s + (0.0957 − 0.0957i)10-s + (−0.707 + 0.707i)11-s + (−0.880 − 0.880i)12-s + 3.32·13-s + (0.127 + 0.127i)14-s − 2.00i·15-s + 3.98·16-s + (−4.00 + 0.994i)17-s + ⋯ |
L(s) = 1 | − 0.0297i·2-s + (−0.254 − 0.254i)3-s + 0.999·4-s + (1.01 + 1.01i)5-s + (−0.00757 + 0.00757i)6-s + (−1.14 + 1.14i)7-s − 0.0595i·8-s − 0.870i·9-s + (0.0302 − 0.0302i)10-s + (−0.213 + 0.213i)11-s + (−0.254 − 0.254i)12-s + 0.921·13-s + (0.0341 + 0.0341i)14-s − 0.517i·15-s + 0.997·16-s + (−0.970 + 0.241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36274 + 0.252294i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36274 + 0.252294i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.707 - 0.707i)T \) |
| 17 | \( 1 + (4.00 - 0.994i)T \) |
good | 2 | \( 1 + 0.0421iT - 2T^{2} \) |
| 3 | \( 1 + (0.440 + 0.440i)T + 3iT^{2} \) |
| 5 | \( 1 + (-2.27 - 2.27i)T + 5iT^{2} \) |
| 7 | \( 1 + (3.03 - 3.03i)T - 7iT^{2} \) |
| 13 | \( 1 - 3.32T + 13T^{2} \) |
| 19 | \( 1 + 5.85iT - 19T^{2} \) |
| 23 | \( 1 + (-3.81 + 3.81i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.616 - 0.616i)T + 29iT^{2} \) |
| 31 | \( 1 + (6.28 + 6.28i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.532 + 0.532i)T + 37iT^{2} \) |
| 41 | \( 1 + (4.20 - 4.20i)T - 41iT^{2} \) |
| 43 | \( 1 + 0.849iT - 43T^{2} \) |
| 47 | \( 1 + 8.83T + 47T^{2} \) |
| 53 | \( 1 - 3.41iT - 53T^{2} \) |
| 59 | \( 1 - 12.4iT - 59T^{2} \) |
| 61 | \( 1 + (-8.41 + 8.41i)T - 61iT^{2} \) |
| 67 | \( 1 - 4.87T + 67T^{2} \) |
| 71 | \( 1 + (2.47 + 2.47i)T + 71iT^{2} \) |
| 73 | \( 1 + (3.10 + 3.10i)T + 73iT^{2} \) |
| 79 | \( 1 + (4.40 - 4.40i)T - 79iT^{2} \) |
| 83 | \( 1 - 6.33iT - 83T^{2} \) |
| 89 | \( 1 - 0.373T + 89T^{2} \) |
| 97 | \( 1 + (-8.46 - 8.46i)T + 97iT^{2} \) |
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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
−12.73971510397303935887994681243, −11.52960961905686885260200204325, −10.83055923890621133474578279953, −9.715479579770720264505729346777, −8.866197641133229630519023230616, −6.85568511051048808713096552625, −6.50430277352873804107862802148, −5.74970345413013094606372624402, −3.17900539535288025155111817514, −2.28448573214436620209050730179,
1.64552615609133119991374495530, 3.53472786914358508296033243486, 5.20729796813615427591745583250, 6.17687421600022848036668822864, 7.19803368530661204521116495757, 8.544689329470373912644933077953, 9.830376978209236452370715204375, 10.48649635326402913736982361851, 11.34567717833949190456032827953, 12.86308525817297096671119640674