L(s) = 1 | − 2·4-s − 7-s + 5·13-s + 4·16-s − 7·19-s − 5·25-s + 2·28-s − 4·31-s + 11·37-s + 8·43-s − 6·49-s − 10·52-s − 61-s − 8·64-s + 5·67-s − 7·73-s + 14·76-s + 17·79-s − 5·91-s − 19·97-s + 10·100-s − 13·103-s + 2·109-s − 4·112-s + ⋯ |
L(s) = 1 | − 4-s − 0.377·7-s + 1.38·13-s + 16-s − 1.60·19-s − 25-s + 0.377·28-s − 0.718·31-s + 1.80·37-s + 1.21·43-s − 6/7·49-s − 1.38·52-s − 0.128·61-s − 64-s + 0.610·67-s − 0.819·73-s + 1.60·76-s + 1.91·79-s − 0.524·91-s − 1.92·97-s + 100-s − 1.28·103-s + 0.191·109-s − 0.377·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5888795834\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5888795834\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−17.54958210256512249523818407718, −16.30001713525188163931408346208, −14.89211076740924954992271086519, −13.58211369833951574831601484907, −12.71563949069013251763955822657, −10.90872829298908742564232271224, −9.429199208210393651255146412199, −8.217650367462526737991465554229, −6.04893540000987436402649531985, −4.04304401379743272242521882180,
4.04304401379743272242521882180, 6.04893540000987436402649531985, 8.217650367462526737991465554229, 9.429199208210393651255146412199, 10.90872829298908742564232271224, 12.71563949069013251763955822657, 13.58211369833951574831601484907, 14.89211076740924954992271086519, 16.30001713525188163931408346208, 17.54958210256512249523818407718