| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s + 7-s − 8-s + 9-s − 2·12-s − 4·13-s − 14-s + 16-s + 6·17-s − 18-s + 2·19-s − 2·21-s + 2·24-s − 5·25-s + 4·26-s + 4·27-s + 28-s − 6·29-s − 4·31-s − 32-s − 6·34-s + 36-s + 2·37-s − 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.436·21-s + 0.408·24-s − 25-s + 0.784·26-s + 0.769·27-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3302236593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3302236593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−19.58002701984274477714555784487, −18.21205413166163302504660601599, −17.21285321618276241317782960612, −16.33708100139526708806159182558, −14.60777306567129604319700846806, −12.30526099005271094386573698884, −11.23136141438460806904855801814, −9.765547119459919407856461234632, −7.57571100088867902110310233811, −5.57928681742950427486583645839,
5.57928681742950427486583645839, 7.57571100088867902110310233811, 9.765547119459919407856461234632, 11.23136141438460806904855801814, 12.30526099005271094386573698884, 14.60777306567129604319700846806, 16.33708100139526708806159182558, 17.21285321618276241317782960612, 18.21205413166163302504660601599, 19.58002701984274477714555784487
