| L(s) = 1 | + 8·3-s − 5·5-s + 14·7-s + 37·9-s − 62·11-s + 42·13-s − 40·15-s − 114·17-s + 70·19-s + 112·21-s − 62·23-s + 25·25-s + 80·27-s − 29·29-s − 142·31-s − 496·33-s − 70·35-s + 146·37-s + 336·39-s + 162·41-s − 352·43-s − 185·45-s + 444·47-s − 147·49-s − 912·51-s − 238·53-s + 310·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s − 0.447·5-s + 0.755·7-s + 1.37·9-s − 1.69·11-s + 0.896·13-s − 0.688·15-s − 1.62·17-s + 0.845·19-s + 1.16·21-s − 0.562·23-s + 1/5·25-s + 0.570·27-s − 0.185·29-s − 0.822·31-s − 2.61·33-s − 0.338·35-s + 0.648·37-s + 1.37·39-s + 0.617·41-s − 1.24·43-s − 0.612·45-s + 1.37·47-s − 3/7·49-s − 2.50·51-s − 0.616·53-s + 0.760·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 29 | \( 1 + p T \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 62 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 70 T + p^{3} T^{2} \) |
| 23 | \( 1 + 62 T + p^{3} T^{2} \) |
| 31 | \( 1 + 142 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 162 T + p^{3} T^{2} \) |
| 43 | \( 1 + 352 T + p^{3} T^{2} \) |
| 47 | \( 1 - 444 T + p^{3} T^{2} \) |
| 53 | \( 1 + 238 T + p^{3} T^{2} \) |
| 59 | \( 1 + 840 T + p^{3} T^{2} \) |
| 61 | \( 1 - 2 T + p^{3} T^{2} \) |
| 67 | \( 1 - 154 T + p^{3} T^{2} \) |
| 71 | \( 1 + 892 T + p^{3} T^{2} \) |
| 73 | \( 1 + 38 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1050 T + p^{3} T^{2} \) |
| 83 | \( 1 - 778 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1410 T + p^{3} T^{2} \) |
| 97 | \( 1 - 466 T + p^{3} T^{2} \) |
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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
−8.114931425690514740569573413149, −7.85343645471940885447680708875, −7.09176589333032254614054280151, −5.86827057516832694327068891082, −4.84423863659621848699717762025, −4.10389421118643447789867394883, −3.17051820039886915576915139737, −2.43362398775419067404814876059, −1.57558170347222304912698906150, 0,
1.57558170347222304912698906150, 2.43362398775419067404814876059, 3.17051820039886915576915139737, 4.10389421118643447789867394883, 4.84423863659621848699717762025, 5.86827057516832694327068891082, 7.09176589333032254614054280151, 7.85343645471940885447680708875, 8.114931425690514740569573413149
