| L(s) = 1 | − 3-s + 5-s − 7-s − 2·9-s − 3·11-s − 13-s − 15-s − 6·17-s − 6·19-s + 21-s + 5·23-s + 25-s + 5·27-s − 6·29-s + 7·31-s + 3·33-s − 35-s − 3·37-s + 39-s + 3·41-s + 2·43-s − 2·45-s + 13·47-s + 49-s + 6·51-s + 4·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.277·13-s − 0.258·15-s − 1.45·17-s − 1.37·19-s + 0.218·21-s + 1.04·23-s + 1/5·25-s + 0.962·27-s − 1.11·29-s + 1.25·31-s + 0.522·33-s − 0.169·35-s − 0.493·37-s + 0.160·39-s + 0.468·41-s + 0.304·43-s − 0.298·45-s + 1.89·47-s + 1/7·49-s + 0.840·51-s + 0.549·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8785343619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8785343619\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.773635570334391101431718781552, −7.78104680479505054359714443551, −6.84477134135160123440672304895, −6.32027464974295412092013963788, −5.55236336259612957808923587214, −4.89929802489774703111507191979, −4.02710370528333780090843591281, −2.73260929560505555562763859686, −2.22764583279416944968103011453, −0.53016725178158614828468649768,
0.53016725178158614828468649768, 2.22764583279416944968103011453, 2.73260929560505555562763859686, 4.02710370528333780090843591281, 4.89929802489774703111507191979, 5.55236336259612957808923587214, 6.32027464974295412092013963788, 6.84477134135160123440672304895, 7.78104680479505054359714443551, 8.773635570334391101431718781552
