| 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
−10.71604992291978735786704847730, −10.31362819800979881614185158886, −9.340498911412799031802533290974, −7.88758641854511586274670860888, −7.13902071408443079066153677310, −6.26906859345727498107098487874, −4.93974866310573931738343063428, −3.90257261300548448075609762739, −2.54160815415552779591803528076, −0.07761952621198305518721249874,
2.33485656548037077746275115247, 3.58612588941893841621277003151, 5.30128442164485013303558407429, 5.88189852552313329998608773840, 6.81656482733590780671920418022, 8.371454291877661228307533513195, 8.926284968727498448994716020691, 9.795833117645334082849209755845, 11.03848935723972375456116410603, 11.94802192613208101474229473289
