L(s) = 1 | − i·3-s − 3.62·7-s − 9-s − 6.20i·11-s + 0.578i·13-s − 1.42·17-s + 5.62i·19-s + 3.62i·21-s − 5.62·23-s + i·27-s + 2i·29-s + 2.57·31-s − 6.20·33-s + 7.83i·37-s + 0.578·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.37·7-s − 0.333·9-s − 1.87i·11-s + 0.160i·13-s − 0.344·17-s + 1.29i·19-s + 0.791i·21-s − 1.17·23-s + 0.192i·27-s + 0.371i·29-s + 0.463·31-s − 1.08·33-s + 1.28i·37-s + 0.0926·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4867042716\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4867042716\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.62T + 7T^{2} \) |
| 11 | \( 1 + 6.20iT - 11T^{2} \) |
| 13 | \( 1 - 0.578iT - 13T^{2} \) |
| 17 | \( 1 + 1.42T + 17T^{2} \) |
| 19 | \( 1 - 5.62iT - 19T^{2} \) |
| 23 | \( 1 + 5.62T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 2.57T + 31T^{2} \) |
| 37 | \( 1 - 7.83iT - 37T^{2} \) |
| 41 | \( 1 - 5.25T + 41T^{2} \) |
| 43 | \( 1 + 7.25iT - 43T^{2} \) |
| 47 | \( 1 - 6.78T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 2.20iT - 59T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 8.41T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 5.42T + 79T^{2} \) |
| 83 | \( 1 - 3.25iT - 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 4.84T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.936175423526271544547350594860, −8.460642773472302202619961707654, −7.61932783802440706564402959468, −6.68239212768573033227118853677, −6.01414255265054616768199200942, −5.66918936784977649085150449847, −4.07344265595133827019981963258, −3.35548540186479242585236426842, −2.53740524689382711968890640842, −1.07728734759227355995901152384,
0.18035925907624018540457071447, 2.13118871086503861692011518589, 2.94652229054856263313718280826, 4.08559552049577346898908031652, 4.58987481826807309364550118405, 5.66788001808844737142903384388, 6.52547095115998231957659661253, 7.13595952867525701155580329340, 7.978742550302285725701720618465, 9.145886679968846785439125403841