| L(s) = 1 | − 2-s − 4-s − 5-s + 4·7-s + 3·8-s + 10-s + 2·11-s − 4·14-s − 16-s − 2·17-s + 6·19-s + 20-s − 2·22-s + 6·23-s + 25-s − 4·28-s − 2·29-s + 10·31-s − 5·32-s + 2·34-s − 4·35-s + 2·37-s − 6·38-s − 3·40-s − 6·41-s + 10·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.51·7-s + 1.06·8-s + 0.316·10-s + 0.603·11-s − 1.06·14-s − 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.223·20-s − 0.426·22-s + 1.25·23-s + 1/5·25-s − 0.755·28-s − 0.371·29-s + 1.79·31-s − 0.883·32-s + 0.342·34-s − 0.676·35-s + 0.328·37-s − 0.973·38-s − 0.474·40-s − 0.937·41-s + 1.52·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.628521543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.628521543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−7.969442651108269867364483714241, −7.42128037066559792393591215373, −6.80492318260472867560284585691, −5.61662059881354107324334466304, −4.90995605920067117221322721145, −4.46271082359165802947890777732, −3.67694005875399616389192429368, −2.53877355899573267136722580164, −1.36388059501191469691228502578, −0.847050927765040751356708691063,
0.847050927765040751356708691063, 1.36388059501191469691228502578, 2.53877355899573267136722580164, 3.67694005875399616389192429368, 4.46271082359165802947890777732, 4.90995605920067117221322721145, 5.61662059881354107324334466304, 6.80492318260472867560284585691, 7.42128037066559792393591215373, 7.969442651108269867364483714241
