L(s) = 1 | + (−0.0494 + 5.65i)2-s − 10.7i·3-s + (−31.9 − 0.559i)4-s + (60.7 + 0.531i)6-s − 198.·7-s + (4.74 − 180. i)8-s + 127.·9-s + 85.9i·11-s + (−6.01 + 343. i)12-s − 407. i·13-s + (9.83 − 1.12e3i)14-s + (1.02e3 + 35.8i)16-s − 1.20e3·17-s + (−6.31 + 721. i)18-s − 206. i·19-s + ⋯ |
L(s) = 1 | + (−0.00874 + 0.999i)2-s − 0.689i·3-s + (−0.999 − 0.0174i)4-s + (0.689 + 0.00602i)6-s − 1.53·7-s + (0.0262 − 0.999i)8-s + 0.524·9-s + 0.214i·11-s + (−0.0120 + 0.689i)12-s − 0.668i·13-s + (0.0134 − 1.53i)14-s + (0.999 + 0.0349i)16-s − 1.01·17-s + (−0.00459 + 0.524i)18-s − 0.130i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0262 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0262 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.108436507\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.108436507\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0494 - 5.65i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 10.7iT - 243T^{2} \) |
| 7 | \( 1 + 198.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 85.9iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 407. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 206. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.19e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.47e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.80e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.26e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.27e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.07e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.13e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 6.26e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.80e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.13e5T + 8.58e9T^{2} \) |
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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
−12.35265041338209929069899859814, −10.58357991049836711230407479296, −9.595555910681156966385512974500, −8.744282573716027072995997134153, −7.37920286978801060959530320323, −6.81040922268458699622637803314, −5.91284299284931694197055397874, −4.46561655381755446742497881354, −3.02399217644644299945849125526, −0.896986888996657402926204561501,
0.46715113521534205448569428590, 2.34382418614193618541868738416, 3.64204675496953805984990614298, 4.40885123498054388830875243454, 5.90665363052762883715784543453, 7.23716120032274693832880598856, 9.077473134212044357738425509257, 9.354338514971821552937877242307, 10.39744223810897048870763052032, 11.12125710809061249248772445789