| L(s) = 1 | − 3-s + 7-s + 9-s + 5·11-s + 2·13-s + 4·17-s + 6·19-s − 21-s − 23-s − 27-s + 29-s + 2·31-s − 5·33-s + 5·37-s − 2·39-s − 7·43-s − 2·47-s + 49-s − 4·51-s − 6·53-s − 6·57-s − 6·59-s + 63-s + 3·67-s + 69-s + 9·71-s − 12·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 0.554·13-s + 0.970·17-s + 1.37·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s + 0.185·29-s + 0.359·31-s − 0.870·33-s + 0.821·37-s − 0.320·39-s − 1.06·43-s − 0.291·47-s + 1/7·49-s − 0.560·51-s − 0.824·53-s − 0.794·57-s − 0.781·59-s + 0.125·63-s + 0.366·67-s + 0.120·69-s + 1.06·71-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.112401659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.112401659\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.300715165083183686251932099930, −7.68986221372037210669022920874, −6.82319275657495445073240050673, −6.21436353994088783168836086831, −5.49007012641082590142659135443, −4.70336095172234251567023074614, −3.84621084418436265943616826552, −3.11253720304210457678203595332, −1.61269602419488992717521044241, −0.955876873948560852752992133608,
0.955876873948560852752992133608, 1.61269602419488992717521044241, 3.11253720304210457678203595332, 3.84621084418436265943616826552, 4.70336095172234251567023074614, 5.49007012641082590142659135443, 6.21436353994088783168836086831, 6.82319275657495445073240050673, 7.68986221372037210669022920874, 8.300715165083183686251932099930
