| L(s) = 1 | − 8·3-s − 6·5-s + 28·7-s + 37·9-s − 24·11-s + 58·13-s + 48·15-s + 17·17-s + 116·19-s − 224·21-s + 60·23-s − 89·25-s − 80·27-s − 30·29-s + 172·31-s + 192·33-s − 168·35-s + 58·37-s − 464·39-s − 342·41-s − 148·43-s − 222·45-s − 288·47-s + 441·49-s − 136·51-s − 318·53-s + 144·55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 0.536·5-s + 1.51·7-s + 1.37·9-s − 0.657·11-s + 1.23·13-s + 0.826·15-s + 0.242·17-s + 1.40·19-s − 2.32·21-s + 0.543·23-s − 0.711·25-s − 0.570·27-s − 0.192·29-s + 0.996·31-s + 1.01·33-s − 0.811·35-s + 0.257·37-s − 1.90·39-s − 1.30·41-s − 0.524·43-s − 0.735·45-s − 0.893·47-s + 9/7·49-s − 0.373·51-s − 0.824·53-s + 0.353·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.323811843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.323811843\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - p T \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 60 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 172 T + p^{3} T^{2} \) |
| 37 | \( 1 - 58 T + p^{3} T^{2} \) |
| 41 | \( 1 + 342 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 + 288 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 252 T + p^{3} T^{2} \) |
| 61 | \( 1 + 110 T + p^{3} T^{2} \) |
| 67 | \( 1 + 484 T + p^{3} T^{2} \) |
| 71 | \( 1 - 708 T + p^{3} T^{2} \) |
| 73 | \( 1 - 362 T + p^{3} T^{2} \) |
| 79 | \( 1 - 484 T + p^{3} T^{2} \) |
| 83 | \( 1 - 756 T + p^{3} T^{2} \) |
| 89 | \( 1 + 774 T + p^{3} T^{2} \) |
| 97 | \( 1 + 382 T + p^{3} T^{2} \) |
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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
−9.724185429600015001815394785506, −8.364663358159391015690911015864, −7.87857168399204373089764919711, −6.91903617208477641196531253629, −5.90691548389614884166499221091, −5.16911693970917115244463594405, −4.62358090297831568925572897784, −3.39834798175105778227790446706, −1.59596755910577983953289919872, −0.70607429953695706186549277016,
0.70607429953695706186549277016, 1.59596755910577983953289919872, 3.39834798175105778227790446706, 4.62358090297831568925572897784, 5.16911693970917115244463594405, 5.90691548389614884166499221091, 6.91903617208477641196531253629, 7.87857168399204373089764919711, 8.364663358159391015690911015864, 9.724185429600015001815394785506
