| L(s) = 1 | − 3·3-s − 5-s + 2·7-s + 6·9-s − 13-s + 3·15-s − 6·21-s − 23-s + 25-s − 9·27-s + 3·29-s − 3·31-s − 2·35-s + 8·37-s + 3·39-s + 3·41-s − 2·43-s − 6·45-s + 11·47-s − 3·49-s + 14·53-s − 8·59-s + 4·61-s + 12·63-s + 65-s − 4·67-s + 3·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 0.755·7-s + 2·9-s − 0.277·13-s + 0.774·15-s − 1.30·21-s − 0.208·23-s + 1/5·25-s − 1.73·27-s + 0.557·29-s − 0.538·31-s − 0.338·35-s + 1.31·37-s + 0.480·39-s + 0.468·41-s − 0.304·43-s − 0.894·45-s + 1.60·47-s − 3/7·49-s + 1.92·53-s − 1.04·59-s + 0.512·61-s + 1.51·63-s + 0.124·65-s − 0.488·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9124687647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9124687647\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.61097695431891812572771540551, −7.23894765301243027339221170097, −6.36358694667981234706594365623, −5.78404075969593536096693616107, −5.12820814876093260820687152704, −4.49132162780431694098830286455, −3.94355360286602801503349744631, −2.60142185594210330236613786748, −1.44280074660581855359414549843, −0.56923778323449802951397178654,
0.56923778323449802951397178654, 1.44280074660581855359414549843, 2.60142185594210330236613786748, 3.94355360286602801503349744631, 4.49132162780431694098830286455, 5.12820814876093260820687152704, 5.78404075969593536096693616107, 6.36358694667981234706594365623, 7.23894765301243027339221170097, 7.61097695431891812572771540551
