| L(s) = 1 | − 3-s + 3·5-s + 7-s + 9-s − 3·11-s − 5·13-s − 3·15-s − 17-s − 2·19-s − 21-s + 6·23-s + 4·25-s − 27-s + 6·29-s − 4·31-s + 3·33-s + 3·35-s − 11·37-s + 5·39-s − 12·41-s + 43-s + 3·45-s + 12·47-s + 49-s + 51-s + 9·53-s − 9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.38·13-s − 0.774·15-s − 0.242·17-s − 0.458·19-s − 0.218·21-s + 1.25·23-s + 4/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.522·33-s + 0.507·35-s − 1.80·37-s + 0.800·39-s − 1.87·41-s + 0.152·43-s + 0.447·45-s + 1.75·47-s + 1/7·49-s + 0.140·51-s + 1.23·53-s − 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.833522216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.833522216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−15.42207980872449, −14.92203515526048, −14.45567649386828, −13.70371609916001, −13.37163968675645, −12.86295021651921, −12.10240954046361, −11.91529239241829, −10.85209851309259, −10.52581930516151, −10.11851327575103, −9.565748792551611, −8.797207540112032, −8.421077686424203, −7.358374266674326, −7.017722335142346, −6.455620745782586, −5.451822638426353, −5.309116693500221, −4.839647813084168, −3.900053289583338, −2.795551929270491, −2.299000609355284, −1.630046970198381, −0.5491579644602730,
0.5491579644602730, 1.630046970198381, 2.299000609355284, 2.795551929270491, 3.900053289583338, 4.839647813084168, 5.309116693500221, 5.451822638426353, 6.455620745782586, 7.017722335142346, 7.358374266674326, 8.421077686424203, 8.797207540112032, 9.565748792551611, 10.11851327575103, 10.52581930516151, 10.85209851309259, 11.91529239241829, 12.10240954046361, 12.86295021651921, 13.37163968675645, 13.70371609916001, 14.45567649386828, 14.92203515526048, 15.42207980872449
