| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s − 3·13-s + 14-s + 16-s − 17-s − 2·19-s + 20-s − 22-s + 4·23-s − 4·25-s + 3·26-s − 28-s + 2·31-s − 32-s + 34-s − 35-s − 7·37-s + 2·38-s − 40-s − 11·43-s + 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.458·19-s + 0.223·20-s − 0.213·22-s + 0.834·23-s − 4/5·25-s + 0.588·26-s − 0.188·28-s + 0.359·31-s − 0.176·32-s + 0.171·34-s − 0.169·35-s − 1.15·37-s + 0.324·38-s − 0.158·40-s − 1.67·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−8.790017458112029102768564254527, −8.045223792528643574322716324258, −7.05123414747805310726255613017, −6.59617030530155453253711877673, −5.63326001475767847139888151345, −4.75009240422348321485231342071, −3.55113833885631977422484227266, −2.54081896456907965748051474419, −1.56474129511226812610105267361, 0,
1.56474129511226812610105267361, 2.54081896456907965748051474419, 3.55113833885631977422484227266, 4.75009240422348321485231342071, 5.63326001475767847139888151345, 6.59617030530155453253711877673, 7.05123414747805310726255613017, 8.045223792528643574322716324258, 8.790017458112029102768564254527
