| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 7-s + 3·8-s + 9-s − 2·11-s − 12-s − 5·13-s − 14-s − 16-s − 4·17-s − 18-s + 21-s + 2·22-s − 4·23-s + 3·24-s − 5·25-s + 5·26-s + 27-s − 28-s + 8·29-s + 3·31-s − 5·32-s − 2·33-s + 4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 1.38·13-s − 0.267·14-s − 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.218·21-s + 0.426·22-s − 0.834·23-s + 0.612·24-s − 25-s + 0.980·26-s + 0.192·27-s − 0.188·28-s + 1.48·29-s + 0.538·31-s − 0.883·32-s − 0.348·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−9.440765134945835622026803358490, −8.660508077875060824408929882116, −7.85631248963382588612315598005, −7.46143359867823390422868408984, −6.16896727785284343819689789988, −4.78782908451205976462170472642, −4.40390657634204558343229968296, −2.85897993273824516683580972698, −1.77743967134967625738915448392, 0,
1.77743967134967625738915448392, 2.85897993273824516683580972698, 4.40390657634204558343229968296, 4.78782908451205976462170472642, 6.16896727785284343819689789988, 7.46143359867823390422868408984, 7.85631248963382588612315598005, 8.660508077875060824408929882116, 9.440765134945835622026803358490
