| L(s) = 1 | + (−0.693 − 0.252i)3-s + (−0.173 − 0.984i)5-s + (−2.32 + 4.03i)7-s + (−1.88 − 1.57i)9-s + (−1.20 − 2.09i)11-s + (−2.11 + 0.768i)13-s + (−0.128 + 0.726i)15-s + (−0.901 + 0.756i)17-s + (−3.98 + 1.77i)19-s + (2.63 − 2.20i)21-s + (−0.255 + 1.44i)23-s + (−0.939 + 0.342i)25-s + (2.01 + 3.48i)27-s + (−3.72 − 3.12i)29-s + (−3.60 + 6.24i)31-s + ⋯ |
| L(s) = 1 | + (−0.400 − 0.145i)3-s + (−0.0776 − 0.440i)5-s + (−0.880 + 1.52i)7-s + (−0.626 − 0.526i)9-s + (−0.364 − 0.630i)11-s + (−0.585 + 0.213i)13-s + (−0.0330 + 0.187i)15-s + (−0.218 + 0.183i)17-s + (−0.913 + 0.406i)19-s + (0.574 − 0.482i)21-s + (−0.0532 + 0.301i)23-s + (−0.187 + 0.0684i)25-s + (0.387 + 0.671i)27-s + (−0.691 − 0.580i)29-s + (−0.647 + 1.12i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0177552 + 0.119664i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0177552 + 0.119664i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (3.98 - 1.77i)T \) |
| good | 3 | \( 1 + (0.693 + 0.252i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (2.32 - 4.03i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.20 + 2.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.11 - 0.768i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.901 - 0.756i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.255 - 1.44i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (3.72 + 3.12i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (3.60 - 6.24i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.81T + 37T^{2} \) |
| 41 | \( 1 + (5.05 + 1.84i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.33 + 7.59i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.70 - 3.94i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.390 + 2.21i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-8.78 + 7.36i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.285 - 1.62i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.80 - 1.51i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.11 - 6.29i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-3.18 - 1.15i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.11 - 0.770i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (5.11 - 8.85i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.26 + 1.91i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-4.19 + 3.52i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.94802192613208101474229473289, −11.03848935723972375456116410603, −9.795833117645334082849209755845, −8.926284968727498448994716020691, −8.371454291877661228307533513195, −6.81656482733590780671920418022, −5.88189852552313329998608773840, −5.30128442164485013303558407429, −3.58612588941893841621277003151, −2.33485656548037077746275115247,
0.07761952621198305518721249874, 2.54160815415552779591803528076, 3.90257261300548448075609762739, 4.93974866310573931738343063428, 6.26906859345727498107098487874, 7.13902071408443079066153677310, 7.88758641854511586274670860888, 9.340498911412799031802533290974, 10.31362819800979881614185158886, 10.71604992291978735786704847730
