| L(s) = 1 | + 3-s + 9-s + 2·11-s − 3·13-s + 6·17-s − 19-s + 4·23-s + 27-s + 10·29-s − 4·31-s + 2·33-s + 3·37-s − 3·39-s + 6·41-s + 4·43-s + 12·47-s + 6·51-s + 6·53-s − 57-s + 8·59-s + 13·61-s + 7·67-s + 4·69-s − 6·71-s − 73-s − 7·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.832·13-s + 1.45·17-s − 0.229·19-s + 0.834·23-s + 0.192·27-s + 1.85·29-s − 0.718·31-s + 0.348·33-s + 0.493·37-s − 0.480·39-s + 0.937·41-s + 0.609·43-s + 1.75·47-s + 0.840·51-s + 0.824·53-s − 0.132·57-s + 1.04·59-s + 1.66·61-s + 0.855·67-s + 0.481·69-s − 0.712·71-s − 0.117·73-s − 0.787·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.692386475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.692386475\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.72759998386731, −12.88301281704263, −12.78779818085880, −12.14617978139910, −11.76205968519094, −11.22018787550372, −10.52867365696021, −10.03165580327865, −9.823792351348233, −9.002825922634319, −8.825033793800807, −8.165120397816530, −7.604504950161688, −7.164085591576242, −6.781456013127345, −5.977753514546984, −5.520393831258997, −4.936891713095289, −4.232946636690754, −3.883943097526061, −3.083966184178347, −2.649501617951794, −2.076810030044006, −1.068994190082534, −0.7673253861121154,
0.7673253861121154, 1.068994190082534, 2.076810030044006, 2.649501617951794, 3.083966184178347, 3.883943097526061, 4.232946636690754, 4.936891713095289, 5.520393831258997, 5.977753514546984, 6.781456013127345, 7.164085591576242, 7.604504950161688, 8.165120397816530, 8.825033793800807, 9.002825922634319, 9.823792351348233, 10.03165580327865, 10.52867365696021, 11.22018787550372, 11.76205968519094, 12.14617978139910, 12.78779818085880, 12.88301281704263, 13.72759998386731
