L(s) = 1 | + 3·5-s + 3·11-s − 2·13-s + 3·17-s + 19-s − 3·23-s + 4·25-s + 6·29-s + 7·31-s − 37-s + 6·41-s − 4·43-s − 9·47-s − 3·53-s + 9·55-s + 9·59-s + 61-s − 6·65-s − 7·67-s + 73-s − 13·79-s + 12·83-s + 9·85-s + 15·89-s + 3·95-s + 10·97-s + 15·101-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 0.904·11-s − 0.554·13-s + 0.727·17-s + 0.229·19-s − 0.625·23-s + 4/5·25-s + 1.11·29-s + 1.25·31-s − 0.164·37-s + 0.937·41-s − 0.609·43-s − 1.31·47-s − 0.412·53-s + 1.21·55-s + 1.17·59-s + 0.128·61-s − 0.744·65-s − 0.855·67-s + 0.117·73-s − 1.46·79-s + 1.31·83-s + 0.976·85-s + 1.58·89-s + 0.307·95-s + 1.01·97-s + 1.49·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.394052064\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.394052064\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.511672581695242399587565478770, −8.608788850096275615758357415082, −7.75445996130784883001003386200, −6.66804808324987664685271338563, −6.17371737327401791885640284583, −5.30404673676025296724252237112, −4.44653635446383902080069146788, −3.21812548500746171890057674885, −2.20240849637471555115239450077, −1.16213428834184120958012944154,
1.16213428834184120958012944154, 2.20240849637471555115239450077, 3.21812548500746171890057674885, 4.44653635446383902080069146788, 5.30404673676025296724252237112, 6.17371737327401791885640284583, 6.66804808324987664685271338563, 7.75445996130784883001003386200, 8.608788850096275615758357415082, 9.511672581695242399587565478770