| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 2·11-s + 5·13-s + 14-s + 16-s − 5·17-s + 8·19-s + 2·22-s − 23-s − 5·26-s − 28-s + 9·29-s + 31-s − 32-s + 5·34-s + 8·37-s − 8·38-s + 2·41-s − 9·43-s − 2·44-s + 46-s + 2·47-s + 49-s + 5·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.603·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 1.21·17-s + 1.83·19-s + 0.426·22-s − 0.208·23-s − 0.980·26-s − 0.188·28-s + 1.67·29-s + 0.179·31-s − 0.176·32-s + 0.857·34-s + 1.31·37-s − 1.29·38-s + 0.312·41-s − 1.37·43-s − 0.301·44-s + 0.147·46-s + 0.291·47-s + 1/7·49-s + 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.403330330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403330330\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−7.999982963847324062647034736069, −6.95689499524609398755937919936, −6.49480533532484019260908332422, −5.82276633065015285800195499587, −5.01473264047083024667181029598, −4.14703272313218731096930964662, −3.20101730470058337400054233775, −2.66229739266779584699943954933, −1.51969568351114257643832157237, −0.66280196744864582583677457574,
0.66280196744864582583677457574, 1.51969568351114257643832157237, 2.66229739266779584699943954933, 3.20101730470058337400054233775, 4.14703272313218731096930964662, 5.01473264047083024667181029598, 5.82276633065015285800195499587, 6.49480533532484019260908332422, 6.95689499524609398755937919936, 7.999982963847324062647034736069
