| L(s) = 1 | − 2·2-s − 3-s + 4·4-s + 7·5-s + 2·6-s − 8·8-s − 26·9-s − 14·10-s + 35·11-s − 4·12-s + 66·13-s − 7·15-s + 16·16-s + 59·17-s + 52·18-s + 137·19-s + 28·20-s − 70·22-s − 7·23-s + 8·24-s − 76·25-s − 132·26-s + 53·27-s + 106·29-s + 14·30-s + 75·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.192·3-s + 1/2·4-s + 0.626·5-s + 0.136·6-s − 0.353·8-s − 0.962·9-s − 0.442·10-s + 0.959·11-s − 0.0962·12-s + 1.40·13-s − 0.120·15-s + 1/4·16-s + 0.841·17-s + 0.680·18-s + 1.65·19-s + 0.313·20-s − 0.678·22-s − 0.0634·23-s + 0.0680·24-s − 0.607·25-s − 0.995·26-s + 0.377·27-s + 0.678·29-s + 0.0852·30-s + 0.434·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.227480730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.227480730\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 - 35 T + p^{3} T^{2} \) |
| 13 | \( 1 - 66 T + p^{3} T^{2} \) |
| 17 | \( 1 - 59 T + p^{3} T^{2} \) |
| 19 | \( 1 - 137 T + p^{3} T^{2} \) |
| 23 | \( 1 + 7 T + p^{3} T^{2} \) |
| 29 | \( 1 - 106 T + p^{3} T^{2} \) |
| 31 | \( 1 - 75 T + p^{3} T^{2} \) |
| 37 | \( 1 - 11 T + p^{3} T^{2} \) |
| 41 | \( 1 + 498 T + p^{3} T^{2} \) |
| 43 | \( 1 - 260 T + p^{3} T^{2} \) |
| 47 | \( 1 + 171 T + p^{3} T^{2} \) |
| 53 | \( 1 + 417 T + p^{3} T^{2} \) |
| 59 | \( 1 + 17 T + p^{3} T^{2} \) |
| 61 | \( 1 - 51 T + p^{3} T^{2} \) |
| 67 | \( 1 - 439 T + p^{3} T^{2} \) |
| 71 | \( 1 + 784 T + p^{3} T^{2} \) |
| 73 | \( 1 - 295 T + p^{3} T^{2} \) |
| 79 | \( 1 + 495 T + p^{3} T^{2} \) |
| 83 | \( 1 - 932 T + p^{3} T^{2} \) |
| 89 | \( 1 + 873 T + p^{3} T^{2} \) |
| 97 | \( 1 + 290 T + p^{3} T^{2} \) |
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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
−13.65686666405865279087515092922, −12.00378766349200127257365368505, −11.32077286063643914702476359855, −10.05716496191669186009209350441, −9.104986172178156877504202881593, −8.067478193358021227618156375693, −6.49956008422671004137457620345, −5.57155480006434920954871171127, −3.30346307076539660289926787276, −1.26004264822092502520571837632,
1.26004264822092502520571837632, 3.30346307076539660289926787276, 5.57155480006434920954871171127, 6.49956008422671004137457620345, 8.067478193358021227618156375693, 9.104986172178156877504202881593, 10.05716496191669186009209350441, 11.32077286063643914702476359855, 12.00378766349200127257365368505, 13.65686666405865279087515092922
