| L(s) = 1 | − 3·3-s − 5·5-s − 20·7-s + 9·9-s − 16·11-s + 58·13-s + 15·15-s + 38·17-s − 4·19-s + 60·21-s + 80·23-s + 25·25-s − 27·27-s + 82·29-s + 8·31-s + 48·33-s + 100·35-s + 426·37-s − 174·39-s − 246·41-s + 524·43-s − 45·45-s + 464·47-s + 57·49-s − 114·51-s − 702·53-s + 80·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.07·7-s + 1/3·9-s − 0.438·11-s + 1.23·13-s + 0.258·15-s + 0.542·17-s − 0.0482·19-s + 0.623·21-s + 0.725·23-s + 1/5·25-s − 0.192·27-s + 0.525·29-s + 0.0463·31-s + 0.253·33-s + 0.482·35-s + 1.89·37-s − 0.714·39-s − 0.937·41-s + 1.85·43-s − 0.149·45-s + 1.44·47-s + 0.166·49-s − 0.313·51-s − 1.81·53-s + 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.130764439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.130764439\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 16 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 38 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 80 T + p^{3} T^{2} \) |
| 29 | \( 1 - 82 T + p^{3} T^{2} \) |
| 31 | \( 1 - 8 T + p^{3} T^{2} \) |
| 37 | \( 1 - 426 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 524 T + p^{3} T^{2} \) |
| 47 | \( 1 - 464 T + p^{3} T^{2} \) |
| 53 | \( 1 + 702 T + p^{3} T^{2} \) |
| 59 | \( 1 - 592 T + p^{3} T^{2} \) |
| 61 | \( 1 - 574 T + p^{3} T^{2} \) |
| 67 | \( 1 - 172 T + p^{3} T^{2} \) |
| 71 | \( 1 + 768 T + p^{3} T^{2} \) |
| 73 | \( 1 + 558 T + p^{3} T^{2} \) |
| 79 | \( 1 + 408 T + p^{3} T^{2} \) |
| 83 | \( 1 + 164 T + p^{3} T^{2} \) |
| 89 | \( 1 + 510 T + p^{3} T^{2} \) |
| 97 | \( 1 - 514 T + p^{3} T^{2} \) |
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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
−11.61767311802924194615316149855, −10.79914480136213803587794293965, −9.880781594903144232539246274705, −8.799158720449665106047987075198, −7.61949523459759201095700230541, −6.51492037772326375473340828650, −5.66222665862326049535392833246, −4.20558789024009583692336867329, −3.01112727604601173818999619452, −0.806760526594359469072808599839,
0.806760526594359469072808599839, 3.01112727604601173818999619452, 4.20558789024009583692336867329, 5.66222665862326049535392833246, 6.51492037772326375473340828650, 7.61949523459759201095700230541, 8.799158720449665106047987075198, 9.880781594903144232539246274705, 10.79914480136213803587794293965, 11.61767311802924194615316149855
