| L(s) = 1 | + 3·3-s − 5·5-s + 16·7-s + 9·9-s − 24·11-s + 14·13-s − 15·15-s − 18·17-s − 36·19-s + 48·21-s + 104·23-s + 25·25-s + 27·27-s + 250·29-s − 28·31-s − 72·33-s − 80·35-s + 54·37-s + 42·39-s + 354·41-s − 228·43-s − 45·45-s + 408·47-s − 87·49-s − 54·51-s − 262·53-s + 120·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.863·7-s + 1/3·9-s − 0.657·11-s + 0.298·13-s − 0.258·15-s − 0.256·17-s − 0.434·19-s + 0.498·21-s + 0.942·23-s + 1/5·25-s + 0.192·27-s + 1.60·29-s − 0.162·31-s − 0.379·33-s − 0.386·35-s + 0.239·37-s + 0.172·39-s + 1.34·41-s − 0.808·43-s − 0.149·45-s + 1.26·47-s − 0.253·49-s − 0.148·51-s − 0.679·53-s + 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.632194522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.632194522\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 14 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 36 T + p^{3} T^{2} \) |
| 23 | \( 1 - 104 T + p^{3} T^{2} \) |
| 29 | \( 1 - 250 T + p^{3} T^{2} \) |
| 31 | \( 1 + 28 T + p^{3} T^{2} \) |
| 37 | \( 1 - 54 T + p^{3} T^{2} \) |
| 41 | \( 1 - 354 T + p^{3} T^{2} \) |
| 43 | \( 1 + 228 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 262 T + p^{3} T^{2} \) |
| 59 | \( 1 - 64 T + p^{3} T^{2} \) |
| 61 | \( 1 + 374 T + p^{3} T^{2} \) |
| 67 | \( 1 + 300 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1016 T + p^{3} T^{2} \) |
| 73 | \( 1 - 274 T + p^{3} T^{2} \) |
| 79 | \( 1 - 788 T + p^{3} T^{2} \) |
| 83 | \( 1 - 396 T + p^{3} T^{2} \) |
| 89 | \( 1 - 786 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1086 T + p^{3} T^{2} \) |
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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
−9.523726154717479476026773019391, −8.622451216424348770286834461342, −8.077545375174310426304560981271, −7.32394268677796378870011129434, −6.31308616316976876108848498693, −5.04208383244018355474991801577, −4.37546954236185352216837165880, −3.20173763802544344705895399962, −2.19207444523359401623467462070, −0.869062758439074154560482522419,
0.869062758439074154560482522419, 2.19207444523359401623467462070, 3.20173763802544344705895399962, 4.37546954236185352216837165880, 5.04208383244018355474991801577, 6.31308616316976876108848498693, 7.32394268677796378870011129434, 8.077545375174310426304560981271, 8.622451216424348770286834461342, 9.523726154717479476026773019391
