| L(s) = 1 | + (−2.42 + 0.883i)2-s + (−0.0430 + 0.243i)3-s + (3.57 − 3.00i)4-s + (0.766 + 0.642i)5-s + (−0.111 − 0.630i)6-s + (0.200 + 0.347i)7-s + (−3.45 + 5.97i)8-s + (2.76 + 1.00i)9-s + (−2.42 − 0.883i)10-s + (−2.59 + 4.49i)11-s + (0.578 + 1.00i)12-s + (0.501 + 2.84i)13-s + (−0.794 − 0.666i)14-s + (−0.189 + 0.159i)15-s + (1.47 − 8.35i)16-s + (3.89 − 1.41i)17-s + ⋯ |
| L(s) = 1 | + (−1.71 + 0.624i)2-s + (−0.0248 + 0.140i)3-s + (1.78 − 1.50i)4-s + (0.342 + 0.287i)5-s + (−0.0453 − 0.257i)6-s + (0.0759 + 0.131i)7-s + (−1.22 + 2.11i)8-s + (0.920 + 0.335i)9-s + (−0.767 − 0.279i)10-s + (−0.782 + 1.35i)11-s + (0.167 + 0.289i)12-s + (0.139 + 0.789i)13-s + (−0.212 − 0.178i)14-s + (−0.0489 + 0.0411i)15-s + (0.368 − 2.08i)16-s + (0.944 − 0.343i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.400893 + 0.319705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.400893 + 0.319705i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (-0.386 + 4.34i)T \) |
| good | 2 | \( 1 + (2.42 - 0.883i)T + (1.53 - 1.28i)T^{2} \) |
| 3 | \( 1 + (0.0430 - 0.243i)T + (-2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (-0.200 - 0.347i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.59 - 4.49i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.501 - 2.84i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.89 + 1.41i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.57 + 2.15i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (6.18 + 2.25i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.13 - 5.42i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.14T + 37T^{2} \) |
| 41 | \( 1 + (-0.496 + 2.81i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (9.52 + 7.99i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (6.35 + 2.31i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-9.42 + 7.90i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-1.42 + 0.518i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (5.35 - 4.49i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.711 - 0.258i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (6.38 + 5.35i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (1.72 - 9.76i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.553 + 3.13i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.75 + 4.77i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.17 + 12.3i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-8.35 + 3.04i)T + (74.3 - 62.3i)T^{2} \) |
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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
−14.78823658644198613437897364870, −13.22442777127968328030280618513, −11.68988481624379382238131067651, −10.36101500847383094081136522219, −9.904660804496007658715704854133, −8.814792186801548102378702485824, −7.41156696002913996905904557011, −6.87486054312476164346994646113, −5.11073742956654168685748009940, −1.99423324995964547440426048028,
1.22278671299037318692833664254, 3.24499215013466444897679070308, 5.91411880362216198900253799576, 7.59469003701862552768954158503, 8.276816424640378868679666848582, 9.572140527649385028067282894622, 10.32208087772534811685331074942, 11.25875133652288195224628156283, 12.47779478806103661500927462064, 13.36218287938082732271219555623
