| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 11-s − 12-s + 13-s + 16-s − 17-s + 18-s + 2·19-s + 22-s + 4·23-s − 24-s + 26-s − 27-s + 2·29-s + 32-s − 33-s − 34-s + 36-s + 6·37-s + 2·38-s − 39-s + 10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.458·19-s + 0.213·22-s + 0.834·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.371·29-s + 0.176·32-s − 0.174·33-s − 0.171·34-s + 1/6·36-s + 0.986·37-s + 0.324·38-s − 0.160·39-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.611700042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.611700042\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.66234863343333, −11.91426110476030, −11.74946383874471, −11.21138167632027, −10.83949152199195, −10.44636274675753, −9.842757044566364, −9.412123596543536, −8.864483676390710, −8.464732210256102, −7.666236216027616, −7.393797845832413, −6.912656730384478, −6.383727494205353, −5.858396732014588, −5.647041388254607, −4.957752462903677, −4.513287930756382, −4.061400869443870, −3.589135583107613, −2.857353325979250, −2.459698934706445, −1.770307118324069, −0.9070266951518944, −0.7198932313583785,
0.7198932313583785, 0.9070266951518944, 1.770307118324069, 2.459698934706445, 2.857353325979250, 3.589135583107613, 4.061400869443870, 4.513287930756382, 4.957752462903677, 5.647041388254607, 5.858396732014588, 6.383727494205353, 6.912656730384478, 7.393797845832413, 7.666236216027616, 8.464732210256102, 8.864483676390710, 9.412123596543536, 9.842757044566364, 10.44636274675753, 10.83949152199195, 11.21138167632027, 11.74946383874471, 11.91426110476030, 12.66234863343333
