| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s − 12-s + 4·13-s + 2·14-s + 16-s + 17-s − 18-s − 4·19-s + 2·21-s + 24-s − 4·26-s − 27-s − 2·28-s + 6·29-s − 4·31-s − 32-s − 34-s + 36-s − 2·37-s + 4·38-s − 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.436·21-s + 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s − 0.328·37-s + 0.648·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8466863892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8466863892\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.886131289758758212352038407617, −8.275039580626643232829872798038, −7.34202395795473137256172457712, −6.47013614452691441044591753850, −6.14933334431625505617815087745, −5.12114461903810821932705460739, −3.99224939529820864221941556336, −3.13517425185144612719079698515, −1.87942099109353383607657594188, −0.66425321809017881628234595837,
0.66425321809017881628234595837, 1.87942099109353383607657594188, 3.13517425185144612719079698515, 3.99224939529820864221941556336, 5.12114461903810821932705460739, 6.14933334431625505617815087745, 6.47013614452691441044591753850, 7.34202395795473137256172457712, 8.275039580626643232829872798038, 8.886131289758758212352038407617
