| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s − 2·13-s + 16-s − 17-s − 18-s + 4·19-s + 22-s − 6·23-s + 24-s − 5·25-s + 2·26-s − 27-s − 3·29-s − 8·31-s − 32-s + 33-s + 34-s + 36-s + 2·37-s − 4·38-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 − 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.213·22-s − 1.25·23-s + 0.204·24-s − 25-s + 0.392·26-s − 0.192·27-s − 0.557·29-s − 1.43·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-s + 0.328·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54978 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54978 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3743237001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3743237001\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.38327332110169, −14.02314844005938, −13.30581125394374, −12.67206055958005, −12.42301852725177, −11.60147266637516, −11.37562881496669, −10.91923427918441, −10.11374775050852, −9.802451129340442, −9.457766025964576, −8.689932771207120, −8.107240177191317, −7.469754484254349, −7.299557538412677, −6.491138132484625, −5.859043291325272, −5.497708620780519, −4.834255243889949, −4.013071959762660, −3.524398513888951, −2.560880832286148, −2.001073314215475, −1.281636390236282, −0.2519018722003008,
0.2519018722003008, 1.281636390236282, 2.001073314215475, 2.560880832286148, 3.524398513888951, 4.013071959762660, 4.834255243889949, 5.497708620780519, 5.859043291325272, 6.491138132484625, 7.299557538412677, 7.469754484254349, 8.107240177191317, 8.689932771207120, 9.457766025964576, 9.802451129340442, 10.11374775050852, 10.91923427918441, 11.37562881496669, 11.60147266637516, 12.42301852725177, 12.67206055958005, 13.30581125394374, 14.02314844005938, 14.38327332110169
