L(s) = 1 | + 3-s + 3·7-s + 9-s − 2·11-s + 3·13-s − 6·17-s + 7·19-s + 3·21-s + 6·23-s + 27-s − 2·29-s + 5·31-s − 2·33-s − 10·37-s + 3·39-s + 12·41-s + 3·43-s − 10·47-s + 2·49-s − 6·51-s + 7·57-s + 6·59-s − 13·61-s + 3·63-s + 7·67-s + 6·69-s + 4·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.832·13-s − 1.45·17-s + 1.60·19-s + 0.654·21-s + 1.25·23-s + 0.192·27-s − 0.371·29-s + 0.898·31-s − 0.348·33-s − 1.64·37-s + 0.480·39-s + 1.87·41-s + 0.457·43-s − 1.45·47-s + 2/7·49-s − 0.840·51-s + 0.927·57-s + 0.781·59-s − 1.66·61-s + 0.377·63-s + 0.855·67-s + 0.722·69-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.339596355\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.339596355\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−9.550056288109099143931230152889, −8.878767058679084531028586330884, −8.137101642137297659611805626833, −7.46821861039588007675945788520, −6.54444166819099512610966331729, −5.29967500393029178357270106346, −4.65945797122965052180088814014, −3.50874630517236808113026847602, −2.43634464161106960470616484114, −1.25044768868847065733824862174,
1.25044768868847065733824862174, 2.43634464161106960470616484114, 3.50874630517236808113026847602, 4.65945797122965052180088814014, 5.29967500393029178357270106346, 6.54444166819099512610966331729, 7.46821861039588007675945788520, 8.137101642137297659611805626833, 8.878767058679084531028586330884, 9.550056288109099143931230152889