L(s) = 1 | − i·2-s − 4-s + 5-s + i·8-s − i·10-s + 4.93i·11-s − 2.49i·13-s + 16-s − 0.433·17-s + 7.93i·19-s − 20-s + 4.93·22-s + 6.02i·23-s + 25-s − 2.49·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.447·5-s + 0.353i·8-s − 0.316i·10-s + 1.48i·11-s − 0.692i·13-s + 0.250·16-s − 0.105·17-s + 1.82i·19-s − 0.223·20-s + 1.05·22-s + 1.25i·23-s + 0.200·25-s − 0.489·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5622524716\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5622524716\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4.93iT - 11T^{2} \) |
| 13 | \( 1 + 2.49iT - 13T^{2} \) |
| 17 | \( 1 + 0.433T + 17T^{2} \) |
| 19 | \( 1 - 7.93iT - 19T^{2} \) |
| 23 | \( 1 - 6.02iT - 23T^{2} \) |
| 29 | \( 1 + 4.71iT - 29T^{2} \) |
| 31 | \( 1 + 5.75iT - 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + 8.73T + 41T^{2} \) |
| 43 | \( 1 + 8.32T + 43T^{2} \) |
| 47 | \( 1 + 5.40T + 47T^{2} \) |
| 53 | \( 1 + 7.27iT - 53T^{2} \) |
| 59 | \( 1 - 3.05T + 59T^{2} \) |
| 61 | \( 1 + 3.10iT - 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 2.47iT - 71T^{2} \) |
| 73 | \( 1 - 5.71iT - 73T^{2} \) |
| 79 | \( 1 + 4.32T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 3.65iT - 97T^{2} \) |
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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
−8.619832074012797862488605254312, −7.926363286815923560511799468938, −7.28086012779033488866525782597, −6.30584946732917145375225957115, −5.50281747116073931419911795257, −4.87314834741831732678167745565, −3.90563150388633748502701495368, −3.23409348615497216831998865974, −2.00633637091141325652434008054, −1.58050931293641078697651764689,
0.14970251639041850358147173280, 1.43693853852274283329739730043, 2.77399699848389249446743844236, 3.48761825089128845727966119396, 4.70454313058538599009838910880, 5.11264536285456829999957029959, 6.08189245927942540525716172491, 6.70912567153978659472475684181, 7.11871319768395305188397110777, 8.342103773283382393322271323094