| L(s) = 1 | − 2·2-s + 4·4-s − 12·5-s − 8·8-s + 24·10-s − 48·11-s − 56·13-s + 16·16-s − 114·17-s − 2·19-s − 48·20-s + 96·22-s + 120·23-s + 19·25-s + 112·26-s + 54·29-s − 236·31-s − 32·32-s + 228·34-s + 146·37-s + 4·38-s + 96·40-s + 126·41-s − 376·43-s − 192·44-s − 240·46-s − 12·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.07·5-s − 0.353·8-s + 0.758·10-s − 1.31·11-s − 1.19·13-s + 1/4·16-s − 1.62·17-s − 0.0241·19-s − 0.536·20-s + 0.930·22-s + 1.08·23-s + 0.151·25-s + 0.844·26-s + 0.345·29-s − 1.36·31-s − 0.176·32-s + 1.15·34-s + 0.648·37-s + 0.0170·38-s + 0.379·40-s + 0.479·41-s − 1.33·43-s − 0.657·44-s − 0.769·46-s − 0.0372·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3305865409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3305865409\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 13 | \( 1 + 56 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 236 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 376 T + p^{3} T^{2} \) |
| 47 | \( 1 + 12 T + p^{3} T^{2} \) |
| 53 | \( 1 + 174 T + p^{3} T^{2} \) |
| 59 | \( 1 - 138 T + p^{3} T^{2} \) |
| 61 | \( 1 + 380 T + p^{3} T^{2} \) |
| 67 | \( 1 + 484 T + p^{3} T^{2} \) |
| 71 | \( 1 + 576 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1150 T + p^{3} T^{2} \) |
| 79 | \( 1 - 776 T + p^{3} T^{2} \) |
| 83 | \( 1 - 378 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1330 T + p^{3} T^{2} \) |
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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
−9.669132110308852985457249800973, −8.869517004996589062169998527504, −7.979087230710626174943351280347, −7.42453833169042217743419287668, −6.65108366495493322458199303175, −5.25154682130264246127995951072, −4.43269485214450475263264893910, −3.09689685651005833588581917795, −2.14138474854096984955876341668, −0.32703630069740052494424668728,
0.32703630069740052494424668728, 2.14138474854096984955876341668, 3.09689685651005833588581917795, 4.43269485214450475263264893910, 5.25154682130264246127995951072, 6.65108366495493322458199303175, 7.42453833169042217743419287668, 7.979087230710626174943351280347, 8.869517004996589062169998527504, 9.669132110308852985457249800973
