| L(s) = 1 | + (0.373 − 0.0794i)2-s + (1.72 − 0.181i)3-s + (−1.69 + 0.754i)4-s + (1.20 + 0.256i)5-s + (0.629 − 0.204i)6-s + (0.104 − 0.994i)7-s + (−1.19 + 0.865i)8-s + (2.93 − 0.623i)9-s + 0.472·10-s + (−3.01 + 1.38i)11-s + (−2.78 + 1.60i)12-s + (−4.33 − 4.80i)13-s + (−0.0399 − 0.379i)14-s + (2.12 + 0.223i)15-s + (2.10 − 2.33i)16-s + (1.5 + 4.61i)17-s + ⋯ |
| L(s) = 1 | + (0.264 − 0.0561i)2-s + (0.994 − 0.104i)3-s + (−0.846 + 0.377i)4-s + (0.540 + 0.114i)5-s + (0.256 − 0.0834i)6-s + (0.0395 − 0.375i)7-s + (−0.421 + 0.305i)8-s + (0.978 − 0.207i)9-s + 0.149·10-s + (−0.908 + 0.417i)11-s + (−0.802 + 0.463i)12-s + (−1.20 − 1.33i)13-s + (−0.0106 − 0.101i)14-s + (0.549 + 0.0577i)15-s + (0.526 − 0.584i)16-s + (0.363 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29922 + 0.0144644i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29922 + 0.0144644i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.72 + 0.181i)T \) |
| 11 | \( 1 + (3.01 - 1.38i)T \) |
| good | 2 | \( 1 + (-0.373 + 0.0794i)T + (1.82 - 0.813i)T^{2} \) |
| 5 | \( 1 + (-1.20 - 0.256i)T + (4.56 + 2.03i)T^{2} \) |
| 7 | \( 1 + (-0.104 + 0.994i)T + (-6.84 - 1.45i)T^{2} \) |
| 13 | \( 1 + (4.33 + 4.80i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 4.61i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (2.30 - 3.99i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.507 + 4.82i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.413 - 0.459i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-4.11 - 2.99i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.264 - 2.51i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (0.927 + 1.60i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-10.9 - 4.85i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-1.26 + 3.88i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.47 + 0.658i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-7.26 + 8.06i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.899 + 2.76i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.118 - 0.0857i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (10.3 - 2.19i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (6.53 - 7.25i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + 6.76T + 89T^{2} \) |
| 97 | \( 1 + (5.86 - 1.24i)T + (88.6 - 39.4i)T^{2} \) |
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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
−13.78570274734942390164774538510, −13.03539481800147395420880231294, −12.34810123411433670828992775998, −10.20167844088658934032947798674, −9.749447626479663857548311397501, −8.199818317609371857041896017372, −7.61311879159866309710557445783, −5.56325286848054776189836955105, −4.11193053311449633801127247473, −2.63884421669695572618177200637,
2.48485563390693407317802539316, 4.35635264462464371282617043427, 5.49419391252283575947147137615, 7.25457108306543526435091046111, 8.699788214578980432527569232617, 9.434054159550483990935825475823, 10.28239897915248852369018601114, 12.11621532646198562605935509505, 13.19724137394970933369055841612, 14.01432334684439153276279893191
