| L(s) = 1 | + 2-s + 2.89·3-s + 4-s + 2.89·6-s − 4.38·7-s + 8-s + 5.38·9-s − 2·11-s + 2.89·12-s − 4.38·14-s + 16-s − 5.86·17-s + 5.38·18-s + 0.973·19-s − 12.6·21-s − 2·22-s − 7.79·23-s + 2.89·24-s + 6.89·27-s − 4.38·28-s + 0.973·29-s + 1.79·31-s + 32-s − 5.79·33-s − 5.86·34-s + 5.38·36-s − 0.591·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.67·3-s + 0.5·4-s + 1.18·6-s − 1.65·7-s + 0.353·8-s + 1.79·9-s − 0.603·11-s + 0.835·12-s − 1.17·14-s + 0.250·16-s − 1.42·17-s + 1.26·18-s + 0.223·19-s − 2.76·21-s − 0.426·22-s − 1.62·23-s + 0.590·24-s + 1.32·27-s − 0.828·28-s + 0.180·29-s + 0.321·31-s + 0.176·32-s − 1.00·33-s − 1.00·34-s + 0.896·36-s − 0.0972·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.89T + 3T^{2} \) |
| 7 | \( 1 + 4.38T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 - 0.973T + 19T^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 - 0.973T + 29T^{2} \) |
| 31 | \( 1 - 1.79T + 31T^{2} \) |
| 37 | \( 1 + 0.591T + 37T^{2} \) |
| 41 | \( 1 + 4.81T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 - 0.381T + 47T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 - 0.973T + 59T^{2} \) |
| 61 | \( 1 + 0.817T + 61T^{2} \) |
| 67 | \( 1 - 1.79T + 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 6.97T + 83T^{2} \) |
| 89 | \( 1 + 0.973T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.48056836847830329810809883224, −6.63400954704746555889226548186, −6.34622676821150826633455054453, −5.27552121438622773729162193973, −4.27492596803446109252984822230, −3.76529279397457404677385463441, −3.04879631076630862538005337193, −2.55211502194329041332436690479, −1.80262589868690942155437903756, 0,
1.80262589868690942155437903756, 2.55211502194329041332436690479, 3.04879631076630862538005337193, 3.76529279397457404677385463441, 4.27492596803446109252984822230, 5.27552121438622773729162193973, 6.34622676821150826633455054453, 6.63400954704746555889226548186, 7.48056836847830329810809883224
