| L(s) = 1 | − 2·3-s + 9-s − 2·11-s − 2·17-s − 2·19-s − 2·23-s + 4·27-s − 6·29-s − 2·31-s + 4·33-s − 6·37-s − 2·41-s − 6·43-s − 8·47-s − 7·49-s + 4·51-s + 2·53-s + 4·57-s − 6·59-s − 14·61-s + 4·69-s − 10·71-s − 2·73-s − 4·79-s − 11·81-s + 12·83-s + 12·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.603·11-s − 0.485·17-s − 0.458·19-s − 0.417·23-s + 0.769·27-s − 1.11·29-s − 0.359·31-s + 0.696·33-s − 0.986·37-s − 0.312·41-s − 0.914·43-s − 1.16·47-s − 49-s + 0.560·51-s + 0.274·53-s + 0.529·57-s − 0.781·59-s − 1.79·61-s + 0.481·69-s − 1.18·71-s − 0.234·73-s − 0.450·79-s − 1.22·81-s + 1.31·83-s + 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.67857495916205, −14.93893620848779, −14.65478150449446, −13.79218141530766, −13.34693872387554, −12.82758349294557, −12.27661439432023, −11.78109197682398, −11.28222009635611, −10.76237165186513, −10.41039892184486, −9.766023365615004, −9.089524081836201, −8.512645503044354, −7.889808677189066, −7.271538776781126, −6.613957845238190, −6.137446876917028, −5.603410395804760, −4.935487290263324, −4.602200417678236, −3.647930879218154, −3.027270720968575, −2.062012235318656, −1.419663469928999, 0, 0,
1.419663469928999, 2.062012235318656, 3.027270720968575, 3.647930879218154, 4.602200417678236, 4.935487290263324, 5.603410395804760, 6.137446876917028, 6.613957845238190, 7.271538776781126, 7.889808677189066, 8.512645503044354, 9.089524081836201, 9.766023365615004, 10.41039892184486, 10.76237165186513, 11.28222009635611, 11.78109197682398, 12.27661439432023, 12.82758349294557, 13.34693872387554, 13.79218141530766, 14.65478150449446, 14.93893620848779, 15.67857495916205
