| L(s) = 1 | − 3-s + 4·5-s + 7-s − 2·9-s − 11-s − 4·15-s + 4·17-s + 2·19-s − 21-s − 7·23-s + 11·25-s + 5·27-s + 8·29-s − 3·31-s + 33-s + 4·35-s + 7·37-s + 7·41-s + 8·43-s − 8·45-s − 3·47-s + 49-s − 4·51-s − 4·55-s − 2·57-s − 6·59-s + 13·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 0.377·7-s − 2/3·9-s − 0.301·11-s − 1.03·15-s + 0.970·17-s + 0.458·19-s − 0.218·21-s − 1.45·23-s + 11/5·25-s + 0.962·27-s + 1.48·29-s − 0.538·31-s + 0.174·33-s + 0.676·35-s + 1.15·37-s + 1.09·41-s + 1.21·43-s − 1.19·45-s − 0.437·47-s + 1/7·49-s − 0.560·51-s − 0.539·55-s − 0.264·57-s − 0.781·59-s + 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.319394806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.319394806\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−14.12857846026953, −13.67148071836649, −13.04934899320723, −12.56464331308396, −12.11734114073089, −11.51188233175427, −11.03262729745784, −10.45977958215426, −10.02744650513997, −9.672358207674391, −9.120686615555139, −8.426081392567615, −8.049597743536653, −7.309695677385048, −6.670855948645158, −5.997782515162648, −5.723860399542525, −5.484636495045347, −4.700991673368593, −4.168349696387796, −3.048608914814749, −2.681931967204016, −2.025183051403977, −1.277823444970927, −0.6554374570646957,
0.6554374570646957, 1.277823444970927, 2.025183051403977, 2.681931967204016, 3.048608914814749, 4.168349696387796, 4.700991673368593, 5.484636495045347, 5.723860399542525, 5.997782515162648, 6.670855948645158, 7.309695677385048, 8.049597743536653, 8.426081392567615, 9.120686615555139, 9.672358207674391, 10.02744650513997, 10.45977958215426, 11.03262729745784, 11.51188233175427, 12.11734114073089, 12.56464331308396, 13.04934899320723, 13.67148071836649, 14.12857846026953
