| L(s) = 1 | + (1 + 2.64i)2-s − 5.29i·3-s + (−6.00 + 5.29i)4-s + (14.0 − 5.29i)6-s + 8·7-s + (−20.0 − 10.5i)8-s − 1.00·9-s − 15.8i·11-s + (28.0 + 31.7i)12-s − 52.9i·13-s + (8 + 21.1i)14-s + (8.00 − 63.4i)16-s + 14·17-s + (−1.00 − 2.64i)18-s − 37.0i·19-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.935i)2-s − 1.01i·3-s + (−0.750 + 0.661i)4-s + (0.952 − 0.360i)6-s + 0.431·7-s + (−0.883 − 0.467i)8-s − 0.0370·9-s − 0.435i·11-s + (0.673 + 0.763i)12-s − 1.12i·13-s + (0.152 + 0.404i)14-s + (0.125 − 0.992i)16-s + 0.199·17-s + (−0.0130 − 0.0346i)18-s − 0.447i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.84329 - 0.457630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84329 - 0.457630i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 - 2.64i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5.29iT - 27T^{2} \) |
| 7 | \( 1 - 8T + 343T^{2} \) |
| 11 | \( 1 + 15.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 52.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 14T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 152T + 1.21e4T^{2} \) |
| 29 | \( 1 + 158. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 224T + 2.97e4T^{2} \) |
| 37 | \( 1 + 243. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 70T + 6.89e4T^{2} \) |
| 43 | \( 1 - 439. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 336T + 1.03e5T^{2} \) |
| 53 | \( 1 + 31.7iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 534. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 95.2iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 174. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 72T + 3.57e5T^{2} \) |
| 73 | \( 1 - 294T + 3.89e5T^{2} \) |
| 79 | \( 1 + 464T + 4.93e5T^{2} \) |
| 83 | \( 1 - 545. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 266T + 7.04e5T^{2} \) |
| 97 | \( 1 + 994T + 9.12e5T^{2} \) |
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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
−12.31042408602567291887373935964, −11.20826438966790886470103959331, −9.747677830036982813410036887934, −8.399307168269159522179731553538, −7.77038059325047720517170938224, −6.80941709861388532632302775140, −5.81546792087560918706530723745, −4.62915306923212696174679123250, −2.94428432024269271217286686958, −0.817983316330969575452015451132,
1.58641848631677491843740680556, 3.29395723250374193763858938729, 4.47297858486166721844959550730, 5.11336679004699689316274306639, 6.77572819989509817789317106527, 8.518527378140111015622430427264, 9.461740363776981184786755855253, 10.19025919520793798687920447140, 11.08060340572120172874678933386, 11.90015358096315189946916544697
