| L(s) = 1 | + 3·3-s − 4·5-s + 26·7-s + 9·9-s − 11·11-s − 32·13-s − 12·15-s + 74·17-s + 60·19-s + 78·21-s + 182·23-s − 109·25-s + 27·27-s − 90·29-s + 8·31-s − 33·33-s − 104·35-s − 66·37-s − 96·39-s + 422·41-s − 408·43-s − 36·45-s + 506·47-s + 333·49-s + 222·51-s + 348·53-s + 44·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.357·5-s + 1.40·7-s + 1/3·9-s − 0.301·11-s − 0.682·13-s − 0.206·15-s + 1.05·17-s + 0.724·19-s + 0.810·21-s + 1.64·23-s − 0.871·25-s + 0.192·27-s − 0.576·29-s + 0.0463·31-s − 0.174·33-s − 0.502·35-s − 0.293·37-s − 0.394·39-s + 1.60·41-s − 1.44·43-s − 0.119·45-s + 1.57·47-s + 0.970·49-s + 0.609·51-s + 0.901·53-s + 0.107·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.758907284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.758907284\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 13 | \( 1 + 32 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 - 60 T + p^{3} T^{2} \) |
| 23 | \( 1 - 182 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 - 8 T + p^{3} T^{2} \) |
| 37 | \( 1 + 66 T + p^{3} T^{2} \) |
| 41 | \( 1 - 422 T + p^{3} T^{2} \) |
| 43 | \( 1 + 408 T + p^{3} T^{2} \) |
| 47 | \( 1 - 506 T + p^{3} T^{2} \) |
| 53 | \( 1 - 348 T + p^{3} T^{2} \) |
| 59 | \( 1 - 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 132 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 71 | \( 1 + 762 T + p^{3} T^{2} \) |
| 73 | \( 1 + 542 T + p^{3} T^{2} \) |
| 79 | \( 1 - 550 T + p^{3} T^{2} \) |
| 83 | \( 1 - 132 T + p^{3} T^{2} \) |
| 89 | \( 1 - 570 T + p^{3} T^{2} \) |
| 97 | \( 1 - 14 T + p^{3} T^{2} \) |
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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
−10.44608256013694293539081591611, −9.486168249201485380695137460830, −8.557304190734888089246808088206, −7.67316838467455093558908240936, −7.28020644243785689545208514319, −5.53472449310190680352534244839, −4.79379423466428761204324213340, −3.61071733649279373374636114573, −2.37278839099267152172120761571, −1.06643353229287896142111462160,
1.06643353229287896142111462160, 2.37278839099267152172120761571, 3.61071733649279373374636114573, 4.79379423466428761204324213340, 5.53472449310190680352534244839, 7.28020644243785689545208514319, 7.67316838467455093558908240936, 8.557304190734888089246808088206, 9.486168249201485380695137460830, 10.44608256013694293539081591611
