| L(s) = 1 | − 3-s − 2·5-s + 4·7-s + 9-s + 13-s + 2·15-s + 2·17-s + 8·19-s − 4·21-s + 8·23-s − 25-s − 27-s − 2·29-s + 4·31-s − 8·35-s − 10·37-s − 39-s + 2·41-s − 4·43-s − 2·45-s − 12·47-s + 9·49-s − 2·51-s + 6·53-s − 8·57-s − 2·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.516·15-s + 0.485·17-s + 1.83·19-s − 0.872·21-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 1.35·35-s − 1.64·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s − 1.75·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s − 1.05·57-s − 0.256·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.145000218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.145000218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−11.51522503864750664764927758570, −11.14536533751795584112384572263, −9.961981705080366148345800006884, −8.660756529929209195819808499841, −7.77831814863968073290767771965, −7.04862524672357691361158313311, −5.43074154894519425045522107114, −4.76628402795015292310147323136, −3.41500334623540265787365604570, −1.28244160692202786359959429283,
1.28244160692202786359959429283, 3.41500334623540265787365604570, 4.76628402795015292310147323136, 5.43074154894519425045522107114, 7.04862524672357691361158313311, 7.77831814863968073290767771965, 8.660756529929209195819808499841, 9.961981705080366148345800006884, 11.14536533751795584112384572263, 11.51522503864750664764927758570
