| L(s) = 1 | + 0.909·2-s + 3-s − 1.17·4-s + 2.40·5-s + 0.909·6-s − 2.88·8-s + 9-s + 2.18·10-s + 0.770·11-s − 1.17·12-s + 0.0110·13-s + 2.40·15-s − 0.276·16-s − 5.24·17-s + 0.909·18-s − 6.56·19-s − 2.81·20-s + 0.701·22-s − 1.61·23-s − 2.88·24-s + 0.770·25-s + 0.0100·26-s + 27-s + 0.735·29-s + 2.18·30-s − 5.78·31-s + 5.51·32-s + ⋯ |
| L(s) = 1 | + 0.642·2-s + 0.577·3-s − 0.586·4-s + 1.07·5-s + 0.371·6-s − 1.02·8-s + 0.333·9-s + 0.690·10-s + 0.232·11-s − 0.338·12-s + 0.00305·13-s + 0.620·15-s − 0.0692·16-s − 1.27·17-s + 0.214·18-s − 1.50·19-s − 0.630·20-s + 0.149·22-s − 0.337·23-s − 0.588·24-s + 0.154·25-s + 0.00196·26-s + 0.192·27-s + 0.136·29-s + 0.398·30-s − 1.03·31-s + 0.975·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 0.909T + 2T^{2} \) |
| 5 | \( 1 - 2.40T + 5T^{2} \) |
| 11 | \( 1 - 0.770T + 11T^{2} \) |
| 13 | \( 1 - 0.0110T + 13T^{2} \) |
| 17 | \( 1 + 5.24T + 17T^{2} \) |
| 19 | \( 1 + 6.56T + 19T^{2} \) |
| 23 | \( 1 + 1.61T + 23T^{2} \) |
| 29 | \( 1 - 0.735T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 - 3.97T + 37T^{2} \) |
| 43 | \( 1 + 3.19T + 43T^{2} \) |
| 47 | \( 1 - 7.16T + 47T^{2} \) |
| 53 | \( 1 + 8.26T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 + 2.45T + 61T^{2} \) |
| 67 | \( 1 - 1.66T + 67T^{2} \) |
| 71 | \( 1 + 1.33T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + 5.21T + 83T^{2} \) |
| 89 | \( 1 - 1.55T + 89T^{2} \) |
| 97 | \( 1 - 1.88T + 97T^{2} \) |
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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
−7.81677548192166164257924743971, −6.77290645281562003130411897167, −6.17005346028278852085123613080, −5.63823095596739992047649003906, −4.56876581656094117165526408587, −4.26553093892515691862382657559, −3.26293190025188399995831565868, −2.38084734185006325142026297890, −1.67193689224286914883052457432, 0,
1.67193689224286914883052457432, 2.38084734185006325142026297890, 3.26293190025188399995831565868, 4.26553093892515691862382657559, 4.56876581656094117165526408587, 5.63823095596739992047649003906, 6.17005346028278852085123613080, 6.77290645281562003130411897167, 7.81677548192166164257924743971
