| L(s) = 1 | + 1.48·2-s − 3-s + 0.193·4-s − 1.48·6-s − 1.19·7-s − 2.67·8-s + 9-s + 11-s − 0.193·12-s − 0.806·13-s − 1.76·14-s − 4.35·16-s − 3.76·17-s + 1.48·18-s − 5.35·19-s + 1.19·21-s + 1.48·22-s − 4·23-s + 2.67·24-s − 1.19·26-s − 27-s − 0.231·28-s − 4.31·29-s + 0.962·31-s − 1.09·32-s − 33-s − 5.58·34-s + ⋯ |
| L(s) = 1 | + 1.04·2-s − 0.577·3-s + 0.0969·4-s − 0.604·6-s − 0.451·7-s − 0.945·8-s + 0.333·9-s + 0.301·11-s − 0.0559·12-s − 0.223·13-s − 0.472·14-s − 1.08·16-s − 0.913·17-s + 0.349·18-s − 1.22·19-s + 0.260·21-s + 0.315·22-s − 0.834·23-s + 0.546·24-s − 0.234·26-s − 0.192·27-s − 0.0437·28-s − 0.800·29-s + 0.172·31-s − 0.193·32-s − 0.174·33-s − 0.957·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 13 | \( 1 + 0.806T + 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4.31T + 29T^{2} \) |
| 31 | \( 1 - 0.962T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 - 9.08T + 41T^{2} \) |
| 43 | \( 1 + 4.41T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 1.42T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 0.0752T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−9.824727548796061474124841653805, −9.083657907431766356766631054875, −8.068013884955445269719119145717, −6.66540924706850267965456337882, −6.28062773744403945913256246318, −5.24767170607294390761269817532, −4.38568330778879800444946357376, −3.61582496401322644909572348923, −2.22506912740385525617645534196, 0,
2.22506912740385525617645534196, 3.61582496401322644909572348923, 4.38568330778879800444946357376, 5.24767170607294390761269817532, 6.28062773744403945913256246318, 6.66540924706850267965456337882, 8.068013884955445269719119145717, 9.083657907431766356766631054875, 9.824727548796061474124841653805
