L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 2·11-s + 4.16·13-s + 14-s + 16-s + 7.32·17-s + 3·19-s + 2·22-s + 1.16·23-s + 4.16·26-s + 28-s + 1.83·29-s − 6.32·31-s + 32-s + 7.32·34-s + 7.48·37-s + 3·38-s + 4·41-s + 3.16·43-s + 2·44-s + 1.16·46-s − 10.4·47-s + 49-s + 4.16·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.377·7-s + 0.353·8-s + 0.603·11-s + 1.15·13-s + 0.267·14-s + 0.250·16-s + 1.77·17-s + 0.688·19-s + 0.426·22-s + 0.242·23-s + 0.816·26-s + 0.188·28-s + 0.341·29-s − 1.13·31-s + 0.176·32-s + 1.25·34-s + 1.23·37-s + 0.486·38-s + 0.624·41-s + 0.482·43-s + 0.301·44-s + 0.171·46-s − 1.52·47-s + 0.142·49-s + 0.577·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.628509094\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.628509094\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4.16T + 13T^{2} \) |
| 17 | \( 1 - 7.32T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 0.162T + 53T^{2} \) |
| 59 | \( 1 + 0.324T + 59T^{2} \) |
| 61 | \( 1 + 3.83T + 61T^{2} \) |
| 67 | \( 1 + 3.48T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 4.32T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.63262370111043843131704370241, −7.01428659592894796935823290869, −6.02262762301723452008359783956, −5.79160217109100923462112886668, −4.93463182735863957646733011930, −4.16643633425528737547236867545, −3.46198825228872491364336373573, −2.90431789479753584336146968192, −1.61730764617821553447878227881, −1.04477543778868374438900366271,
1.04477543778868374438900366271, 1.61730764617821553447878227881, 2.90431789479753584336146968192, 3.46198825228872491364336373573, 4.16643633425528737547236867545, 4.93463182735863957646733011930, 5.79160217109100923462112886668, 6.02262762301723452008359783956, 7.01428659592894796935823290869, 7.63262370111043843131704370241