| L(s) = 1 | + (−0.997 − 0.0697i)2-s + (−0.0352 − 1.00i)3-s + (0.990 + 0.139i)4-s + (−0.727 + 2.11i)5-s + (−0.0352 + 1.00i)6-s + (−1.27 − 2.20i)7-s + (−0.978 − 0.207i)8-s + (1.97 − 0.138i)9-s + (0.872 − 2.05i)10-s + (0.431 + 4.10i)11-s + (0.105 − 1.00i)12-s + (−1.81 − 2.68i)13-s + (1.11 + 2.29i)14-s + (2.15 + 0.659i)15-s + (0.961 + 0.275i)16-s + (0.925 − 2.29i)17-s + ⋯ |
| L(s) = 1 | + (−0.705 − 0.0493i)2-s + (−0.0203 − 0.582i)3-s + (0.495 + 0.0695i)4-s + (−0.325 + 0.945i)5-s + (−0.0143 + 0.411i)6-s + (−0.481 − 0.833i)7-s + (−0.345 − 0.0735i)8-s + (0.658 − 0.0460i)9-s + (0.276 − 0.650i)10-s + (0.130 + 1.23i)11-s + (0.0304 − 0.289i)12-s + (−0.502 − 0.745i)13-s + (0.298 + 0.612i)14-s + (0.557 + 0.170i)15-s + (0.240 + 0.0689i)16-s + (0.224 − 0.555i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01965 + 0.0789995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01965 + 0.0789995i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.997 + 0.0697i)T \) |
| 5 | \( 1 + (0.727 - 2.11i)T \) |
| 19 | \( 1 + (-0.663 - 4.30i)T \) |
| good | 3 | \( 1 + (0.0352 + 1.00i)T + (-2.99 + 0.209i)T^{2} \) |
| 7 | \( 1 + (1.27 + 2.20i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.431 - 4.10i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (1.81 + 2.68i)T + (-4.86 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-0.925 + 2.29i)T + (-12.2 - 11.8i)T^{2} \) |
| 23 | \( 1 + (-1.92 - 1.85i)T + (0.802 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-2.43 - 6.03i)T + (-20.8 + 20.1i)T^{2} \) |
| 31 | \( 1 + (-4.46 + 4.95i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-8.23 - 5.97i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.18 + 0.339i)T + (34.7 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.951 - 5.39i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.92 - 9.72i)T + (-33.8 + 32.6i)T^{2} \) |
| 53 | \( 1 + (-6.14 - 0.864i)T + (50.9 + 14.6i)T^{2} \) |
| 59 | \( 1 + (2.88 + 11.5i)T + (-52.0 + 27.6i)T^{2} \) |
| 61 | \( 1 + (0.716 + 0.691i)T + (2.12 + 60.9i)T^{2} \) |
| 67 | \( 1 + (-3.67 - 2.29i)T + (29.3 + 60.2i)T^{2} \) |
| 71 | \( 1 + (-7.58 - 4.03i)T + (39.7 + 58.8i)T^{2} \) |
| 73 | \( 1 + (-3.99 + 5.92i)T + (-27.3 - 67.6i)T^{2} \) |
| 79 | \( 1 + (-0.254 - 7.27i)T + (-78.8 + 5.51i)T^{2} \) |
| 83 | \( 1 + (2.69 - 2.99i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-15.3 + 4.41i)T + (75.4 - 47.1i)T^{2} \) |
| 97 | \( 1 + (-5.61 + 3.50i)T + (42.5 - 87.1i)T^{2} \) |
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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
−9.872571575051666957550098001645, −9.712596892749215218607461994826, −7.962297186451354533046301138928, −7.55031333192172290718047161808, −6.92980149978596020365906197893, −6.24875097993108774743089335590, −4.65941619621606747025963776444, −3.51364459784020129979455066108, −2.43526811215355043178550611846, −1.03511472955124023021014039862,
0.791752963305659428150827705848, 2.41354257929582885180613659446, 3.73652416313453962371603937862, 4.73798527293707159895148834790, 5.70242387581222390989318914126, 6.64407900830430382961880347191, 7.69710566982157130724693120212, 8.790740305344241109016467482726, 8.971909645116931542835422574236, 9.840012082739644558986906142219
