L(s) = 1 | + 2-s − i·3-s + 4-s − i·6-s + 8-s − 9-s − 4i·11-s − i·12-s + (2 − 3i)13-s + 16-s − 4i·17-s − 18-s + 4i·19-s − 4i·22-s + 6i·23-s − i·24-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.5·4-s − 0.408i·6-s + 0.353·8-s − 0.333·9-s − 1.20i·11-s − 0.288i·12-s + (0.554 − 0.832i)13-s + 0.250·16-s − 0.970i·17-s − 0.235·18-s + 0.917i·19-s − 0.852i·22-s + 1.25i·23-s − 0.204i·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.522449936\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.522449936\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-2 + 3i)T \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−8.886523840401281914411206428382, −7.84954046133758381871875453885, −7.57890789636835864850367380507, −6.32348519188408125550787316452, −5.82462603755263349736329311202, −5.14582123041252896236355918626, −3.76623472439731720679553981675, −3.21226331271031054578421884867, −2.04031592071084337002022513943, −0.71478162549208498852725680244,
1.61752912988400627581928835745, 2.69536341333025241394149460749, 3.79297305775890104753365929140, 4.53127134664982239360982595946, 5.08255065135544625293200762185, 6.32264056210951013726041810822, 6.73097409306985938747973742310, 7.78903191326942455805947105579, 8.696552147254585597012943373384, 9.434667563293574858454104998240